
Citation: Marcos Morey, Ana Fernández-Marmiesse, Jose Angel Cocho, María L. Couce. Influence of technology in genetic epidemiology[J]. AIMS Genetics, 2015, 2(3): 219-229. doi: 10.3934/genet.2015.3.219
[1] | Yantao Song, Wenjie Zhang, Yue Zhang . A novel lightweight deep learning approach for simultaneous optic cup and optic disc segmentation in glaucoma detection. Mathematical Biosciences and Engineering, 2024, 21(4): 5092-5117. doi: 10.3934/mbe.2024225 |
[2] | Rafsanjany Kushol, Md. Hasanul Kabir, M. Abdullah-Al-Wadud, Md Saiful Islam . Retinal blood vessel segmentation from fundus image using an efficient multiscale directional representation technique Bendlets. Mathematical Biosciences and Engineering, 2020, 17(6): 7751-7771. doi: 10.3934/mbe.2020394 |
[3] | Yue Li, Hongmei Jin, Zhanli Li . A weakly supervised learning-based segmentation network for dental diseases. Mathematical Biosciences and Engineering, 2023, 20(2): 2039-2060. doi: 10.3934/mbe.2023094 |
[4] | Jianguo Xu, Cheng Wan, Weihua Yang, Bo Zheng, Zhipeng Yan, Jianxin Shen . A novel multi-modal fundus image fusion method for guiding the laser surgery of central serous chorioretinopathy. Mathematical Biosciences and Engineering, 2021, 18(4): 4797-4816. doi: 10.3934/mbe.2021244 |
[5] | Dehua Feng, Xi Chen, Xiaoyu Wang, Xuanqin Mou, Ling Bai, Shu Zhang, Zhiguo Zhou . Predicting effectiveness of anti-VEGF injection through self-supervised learning in OCT images. Mathematical Biosciences and Engineering, 2023, 20(2): 2439-2458. doi: 10.3934/mbe.2023114 |
[6] | Ran Zhou, Yanghan Ou, Xiaoyue Fang, M. Reza Azarpazhooh, Haitao Gan, Zhiwei Ye, J. David Spence, Xiangyang Xu, Aaron Fenster . Ultrasound carotid plaque segmentation via image reconstruction-based self-supervised learning with limited training labels. Mathematical Biosciences and Engineering, 2023, 20(2): 1617-1636. doi: 10.3934/mbe.2023074 |
[7] | Duolin Sun, Jianqing Wang, Zhaoyu Zuo, Yixiong Jia, Yimou Wang . STS-TransUNet: Semi-supervised Tooth Segmentation Transformer U-Net for dental panoramic image. Mathematical Biosciences and Engineering, 2024, 21(2): 2366-2384. doi: 10.3934/mbe.2024104 |
[8] | Zhenwu Xiang, Qi Mao, Jintao Wang, Yi Tian, Yan Zhang, Wenfeng Wang . Dmbg-Net: Dilated multiresidual boundary guidance network for COVID-19 infection segmentation. Mathematical Biosciences and Engineering, 2023, 20(11): 20135-20154. doi: 10.3934/mbe.2023892 |
[9] | Wenli Cheng, Jiajia Jiao . An adversarially consensus model of augmented unlabeled data for cardiac image segmentation (CAU+). Mathematical Biosciences and Engineering, 2023, 20(8): 13521-13541. doi: 10.3934/mbe.2023603 |
[10] | Jingyao Liu, Qinghe Feng, Yu Miao, Wei He, Weili Shi, Zhengang Jiang . COVID-19 disease identification network based on weakly supervised feature selection. Mathematical Biosciences and Engineering, 2023, 20(5): 9327-9348. doi: 10.3934/mbe.2023409 |
To exemplify the phenomena of compounds scientifically, researchers utilize the contraption of the diagrammatic hypothesis, it is a well-known branch of geometrical science named graph theory. This division of numerical science provides its services in different fields of sciences. The particular example in networking [1], from electronics [2], and for the polymer industry, we refer to see [3]. Particularly in chemical graph theory, this division has extra ordinary assistance to study giant and microscope-able chemical compounds. For such a study, researchers made some transformation rules to transfer a chemical compound to a discrete pattern of shapes (graph). Like, an atom represents as a vertex and the covalent bonding between atoms symbolized as edges. Such transformation is known as molecular graph theory. A major importance of this alteration is that the hydrogen atoms are omitted. Some chemical structures and compounds conversion are presented in [4,5,6].
In cheminformatics, the topological index gains attraction due to its implementations. Various topological indices help to estimate a bio-activity and physicochemical characteristics of a chemical compound. Some interesting and useful topological indices for various chemical compounds are studied in [3,7,8]. A topological index modeled a molecular graph or a chemical compound into a numerical value. Since 1947, topological index implemented in chemistry [9], biology [10], and information science [11,12]. Sombor index and degree-related properties of simplicial networks [13], Nordhaus–Gaddum-type results for the Steiner Gutman index of graphs [14], Lower bounds for Gaussian Estrada index of graphs [15], On the sum and spread of reciprocal distance Laplacian eigenvalues of graphs in terms of Harary index [16], the expected values for the Gutman index, Schultz index, and some Sombor indices of a random cyclooctane chain [17,18,19], bounds on the partition dimension of convex polytopes [20,21], computing and analyzing the normalized Laplacian spectrum and spanning tree of the strong prism of the dicyclobutadieno derivative of linear phenylenes [22], on the generalized adjacency, Laplacian and signless Laplacian spectra of the weighted edge corona networks [23,24], Zagreb indices and multiplicative Zagreb indices of Eulerian graphs [25], Minimizing Kirchhoff index among graphs with a given vertex bipartiteness, [26], asymptotic Laplacian energy like invariant of lattices [27]. Few interesting studies regarding the chemical graph theory can be found in [28,29,30,31,32].
Recently, the researchers of [33] introduced a topological descriptor and called the face index. Moreover, the idea of computing structure-boiling point and energy of a structure, motivated them to introduced this parameter without heavy computation. They computed these parameters for different models compare the results with previous literature and found approximate solutions with comparatively less computations. This is all the blessings of face index of a graph. The major concepts of this research work are elaborated in the given below definitions.
Definition 1.1. [33] Let a graph $ G = \left(V(G), E(G), F(G)\right) $ having face, edge and vertex sets notation with $ F(G), E(G), V(G), $ respectively. It is mandatory that the graph is connected, simple and planar. If $ e $ from the edge set $ E(G), $ is one of those edges which surrounds a face, then the face $ f $ from the face set $ F(G), $ is incident to the edge $ e. $ Likewise, if a vertex $ \alpha $ from the vertex set $ V(G) $ is at the end of those incident edges, then a face $ f $ is incident to that vertex. This face-vertex incident relation is symbolized here by the notation $ \alpha\sim f. $ The face degree of $ f $ in $ G $ is described as $ d(f) = \sum_{\alpha\sim f}{d\left(\alpha\right)}, $ which are elaborated in the Figure 1.
Definition 1.2. [33] The face index $ FI\left(G\right), $ for a graph $ G, $ is formulated as
$ FI(G)=∑f∈F(G)d(f)=∑α∼f,f∈F(G)d(α). $
|
In the Figure 1, we can see that there are two faces with degree $ 4, $ exactly two with five count and four with count of 6. Moreover, there is an external face with count of face degree $ 28, $ which is the count of vertices.
As the information given above that the face index is quite new and introduced in the year 2020, so there is not so much literature is available. A few recent studies on this topic are summarized here. A chemical compound of silicon carbides is elaborated with such novel definition in [34]. Some carbon nanotubes are discussed in [35]. Except for the face index, there are distance and degree-based graphical descriptors available in the literature. For example, distance-based descriptors of phenylene nanotube are studied in [36], and in [37] titania nanotubes are discussed with the same concept. Star networks are studied in [38], with the concept of degree-based descriptors. Bounds on the descriptors of some generalized graphs are discussed in [39]. General Sierpinski graph is discussed in [40], in terms of different topological descriptor aspects. The study of hyaluronic acid-doxorubicin ar found in [41], with the same concept of the index. The curvilinear regression model of the topological index for the COVID-19 treatment is discussed in [42]. For further reading and interesting advancements of topological indices, polynomials of zero-divisor structures are found in [43], zero divisor graph of commutative rings [44], swapped networks modeled by optical transpose interconnection system [45], metal trihalides network [46], some novel drugs used in the cancer treatment [47], para-line graph of Remdesivir used in the prevention of corona virus [48], tightest nonadjacently configured stable pentagonal structure of carbon nanocones [49]. In order to address a novel preventive category (P) in the HIV system known as the HIPV mathematical model, the goal of this study is to offer a design of a Morlet wavelet neural network (MWNN) [50].
In the next section, we discussed the newly developed face index or face-based index for different chemical compounds. Silicate network, triangular honeycomb network, carbon sheet, polyhedron generalized sheet, and generalized chain of silicate network are studied with the concept of the face-based index. Given that the face index is more versatile than vertex degree-based topological descriptors, this study will aid in understanding the structural characteristics of chemical networks. Only the difficulty authors will face to compute the face degree of a generalized network or structure, because it is more generalized version and taking degree based partition of edges into this umbrella of face index.
Silicates are formed when metal carbonates or metal oxides react with sand. The $ SiO_4, $ which has a tetrahedron structure, is the fundamental chemical unit of silicates. The central vertex of the $ SiO_4 $ tetrahedron is occupied by silicon ions, while the end vertices are occupied by oxygen ions [51,52,53]. A silicate sheet is made up of rings of tetrahedrons that are joined together in a two-dimensional plane by oxygen ions from one ring to the other to form a sheet-like structure. The silicate network $ SL_{n} $ symbol, where $ {n} $ represents the total number of hexagons occurring between the borderline and center of the silicate network $ SL_{n}. $ The silicate network of dimension one is depicted in Figure 2. It contain total $ 3{n}\left(5{n}+1\right) $ vertices are $ 36{n}^2 $ edges. Moreover, the result required is detailed are available in Table 1.
Dimension | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{12}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{15}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{36}}}{\bf{|}} $ |
$ 1 $ | $ 24 $ | $ 48 $ | $ 7 $ |
$ 2 $ | $ 32 $ | $ 94 $ | $ 14 $ |
$ 3 $ | $ 40 $ | $ 152 $ | $ 23 $ |
$ 4 $ | $ 48 $ | $ 222 $ | $ 34 $ |
$ 5 $ | $ 56 $ | $ 304 $ | $ 47 $ |
$ 6 $ | $ 64 $ | $ 398 $ | $ 62 $ |
$ 7 $ | $ 72 $ | $ 504 $ | $ 79 $ |
$ 8 $ | $ 80 $ | $ 622 $ | $ 98 $ |
. | . | . | . |
. | . | . | . |
. | . | . | . |
$ {n} $ | $ 8{n}+16 $ | $ 6{n}^{2}+28{n}+14 $ | $ {n}^{2}+4{n}+2 $ |
Theorem 2.1. Let $ SL_{n} $ be the silicate network of dimension $ n\geq1. $ Then the face index of $ SL_{n} $ is
$ FI(SLn)=126n2+720n+558. $
|
Proof. Consider $ SL_{n} $ the graph of silicate network with dimension $ {n}. $ Suppose $ f_{i} $ denotes the faces of graph $ SL_{n} $ having degree $ i. $ that is, $ d(f_{i}) = \sum_{\alpha\sim f_{i}}{d\left(\alpha\right)} = i $ and $ |f_{i}| $ denotes the number of faces with degree $ i. $ The graph $ SL_{n} $ contains three types of internal faces $ f_{12}, $ $ f_{15}, $ $ f_{36}, $ and single external face which is usually denoted by $ f^{\infty}. $
If $ SL_{n} $ has one dimension then sum of degree of vertices incident to the external face is $ 144 $ and when $ SL_{n} $ has two dimension then sum of degree of incident vertices to the external face is $ 204 $ whenever $ SL_{n} $ has three dimension then sum of degree of incident vertices to the external face is $ 264. $ Similarly, $ SL_{n} $ has $ n- $dimension then sum of degree of incident vertices to the external face is $ 60n+84. $
The number of internal faces with degree in each dimension is mentioned in Table 1.
By using the definition of face index $ FI $ we have
$ FI(SLn)=∑α∼f∈F(SLn)d(α)=∑α∼f12∈F(SLn)d(α)+∑α∼f15∈F(SLn)d(α)+∑α∼f36∈F(SLn)d(α)+∑α∼f∞∈F(SLn)d(α)=|f12|(12)+|f15|(15)+|f36|(36)+(60n+84)=(8n+16)(12)+(6n2+28n+14)(15)+(n2+4n+2)(36)+60n+84=126n2+72n+558. $
|
Hence, this is our required result.
A chain silicate network of dimension $ (m, n) $ is symbolized as $ CSL\left(m, n\right) $ which is made by arranging $ (m, n) $ tetrahedron molecules linearly. A chain silicate network of dimension $ (m, n) $ with $ m, n\geq1 $ where $ m $ denotes the number of rows and each row has $ n $ number of tetrahedrons. The following theorem formulates the face index $ FI $ for chain silicate network.
Theorem 2.2. Let $ CSL\left(m, n\right) $ be the chain of silicate network of dimension $ m, n\geq1. $ Then the face index $ FI $ of the graph $ CSL\left(m, n\right) $ is
$ FI(CSL(m,n))={48n−12if m=1, n≥1;96m−12if n=1, m≥2;168m−60if n=2,m≥2;45m−9n+36mn−42if both m,n are even45m−9n+36mn−21otherwise. $
|
Proof. Let $ CSL\left(m, n\right) $ be the graph of chain silicate network of dimension $ (m, n) $ with $ m, n\geq1 $ where $ m $ represents the number of rows and $ n $ is the number of tetrahedrons in each row. A graph $ CSL\left(m, n\right) $ for $ m = 1 $ contains three type of internal faces $ f_{9}, \ f_{12} $ and $ f_{15} $ with one external face $ f^{\infty}. $ While for $ m\geq2, $ it has four type of internal faces $ f_{9}, \ f_{12}, \ f_{15} $ and $ f_{36} $ with one external face $ f^{\infty}. $ We want to evaluate the algorithm of face index $ FI $ for chain silicate network. We will discuss it in two different cases.
Case 1: When $ CSL\left(m, n\right) $ has one row $ (m = 1) $ with $ n $ number of tetrahedrons as shown in the Figure 3.
The graph has three type of internal faces $ f_{9}, \ f_{12} $ and $ f_{15} $ with one external face $ f^{\infty}. $ The sum of degree of incident vertices to the external face is $ 9n $ and number of faces are $ |f_{9}| = 2, \ |f_{12}| = 2n $ and $ |f_{15}| = n-2. $ Now the face index $ FI $ of the graph $ CSL\left(m, n\right) $ is given by
$ FI(CSL(m,n))=∑α∼f∈F(CSL(m,n))d(α)=∑α∼f9∈F(CSL(m,n))d(α)+∑α∼f12∈F(CSL(m,n))d(α)+∑α∼f15∈F(CSL(m,n))d(α)+∑α∼f∞∈F(CSL(m,n))d(α)=|f9|(9)+|f12|(12)+|f15|(15)+(9n)=(2)(9)+(2n)(12)+(n−2)(15)+9n=48n−12. $
|
Case 2: When $ CSL\left(m, n\right) $ has more than one rows $ (m\neq1) $ with $ n $ number of tetrahedrons in each row as shown in the Figure 4.
The graph has four type of internal faces $ f_{9}, \ f_{12}, \ f_{15} $ and $ f_{36} $ with one external face $ f^{\infty}. $ The sum of degree of incident vertices to the external face is
$ ∑α∼f∞∈F(CSL(m,n))d(α)={18mif n=1, m≥1;27mif n=2, m≥1;30m+15n−30if both m,n are even30m+15n−33otherwise. $
|
The number of faces are $ |f_{9}|, \ |f_{12}|, \ f_{15} $ and $ |f_{36}| $ are given by
$ |f9|={2if m is odd3+(−1)nif m is even.|f12|={2(2m+n−1)if m is odd4(⌊n+12⌋+2m−1)if m is even|f15|=(3m−2)n−m|f36|={(m−12)(n−1)if m is odd(2n+(−1)n−14)(m−22)nif m is even. $
|
Now the face index $ FI $ of the graph $ CSL\left(m, n\right) $ is given by
$ FI(CSL(m,n))=∑α∼f∈F(CSL(m,n))d(α)=∑α∼f9∈F(CSL(m,n))d(α)+∑α∼f12∈F(CSL(m,n))d(α)+∑α∼f15∈F(CSL(m,n))d(α)+∑α∼f36∈F(CSL(m,n))d(α)+∑α∼f∞∈F(CSL(m,n))d(α)=|f9|(9)+|f12|(12)+|f15|(15)+|f36|(36)+∑α∼f∞∈F(CSL(m,n))d(α). $
|
After some mathematical simplifications, we can get
$ FI(CSL(m,n))={48n−12if m=196m−12if n=1,∀m168m−60if n=2,∀m45m−9n+36mn−42if both m,n are even45m−9n+36mn−21otherwise. $
|
There are three regular plane tessellations known to exist, each constituted from the same type of regular polygon: triangular, square, and hexagonal. The triangular tessellation is used to define the hexagonal network, which is extensively studied in [54]. A dimensioned hexagonal network $ TH_{k} $ has $ 3{k}^2-3{k}+1 $ vertices and $ 9{k}^2-15{k}+6 $ edges, where $ {k} $ is the number of vertices on one side of the hexagon. It has $ 2{k}-2 $ diameter. There are six vertices of degree three that are referred to as corner vertices. Moreover, the result required detailed are available in the Table 2.
Dimension | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{12}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{14}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{17}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{18}}}{\bf{|}} $ |
$ 1 $ | $ 6 $ | $ 0 $ | $ 0 $ | $ 0 $ |
$ 2 $ | $ 6 $ | $ 12 $ | $ 12 $ | $ 12 $ |
$ 3 $ | $ 6 $ | $ 24 $ | $ 24 $ | $ 60 $ |
$ 4 $ | $ 6 $ | $ 36 $ | $ 36 $ | $ 144 $ |
$ 5 $ | $ 6 $ | $ 48 $ | $ 48 $ | $ 264 $ |
$ 6 $ | $ 6 $ | $ 60 $ | $ 60 $ | $ 420 $ |
$ 7 $ | $ 6 $ | $ 72 $ | $ 72 $ | $ 612 $ |
$ 8 $ | $ 6 $ | $ 84 $ | $ 84 $ | $ 840 $ |
. | . | . | . | . |
. | . | . | . | . |
. | . | . | . | . |
$ {k} $ | $ 6 $ | $ 12({k}-1) $ | $ 12({k}-1) $ | $ 18{k}^{2}-42{k}+24 $ |
Theorem 2.3. Let $ TH_{k} $ be the triangular honeycomb network of dimension $ k\geq1. $ Then the face index of graph $ TH_{k} $ is
$ FI(THk)=324k2−336k+102. $
|
Proof. Consider $ TH_{k} $ be a graph of triangular honeycomb network. The graph $ TH_{1} $ has one internal and only one external face while graph $ TH_{k} $ with $ k\geq2, $ contains four types of internal faces $ f_{12}, $ $ f_{14}, $ $ f_{17}, $ and $ f_{18} $ with one external face $ f^{\infty}. $
For $ TH_{1} $ the sum of degree of incident vertices to the external face is $ 18 $ and in $ TH_{2} $ the sum of degree of incident vertices to the external face is $ 66. $ Whenever the graph $ TH_{3}, $ the sum of degree of incident vertices to the external face is $ 114. $ Similarly, for $ TH_{k} $ has $ n- $dimension then sum of degree of incident vertices to the external face is $ 48k-30. $
The number of internal faces with degree in each dimension is given in Table 2.
By using the definition of face index $ FI $ we have
$ FI(THk)=∑α∼f∈F(THk)d(α)=∑α∼f12∈F(THk)d(α)+∑α∼f14∈F(THk)d(α)+∑α∼f17∈F(THk)d(α)+∑α∼f18∈F(THk)d(α)+∑α∼f∞∈F(THk)d(α)=|f12|(12)+|f14|(14)+|f17|(17)+|f18|(18)+(48k−30)=(6)(12)+(12(k−1))(14)+(12(k−1))(17)+(18k2−42k+24)(18)+48k−30=324k2−336k+102. $
|
Hence, this is our required result.
Given carbon sheet in the Figure 6, is made by grid of hexagons. There are few types of carbon sheets are given in [55,56]. The carbon sheet is symbolize as $ HCS_{{m}, {n}}, $ where $ {n} $ represents the total number of vertical hexagons and $ {m} $ denotes the horizontal hexagons. It contain total $ 4{m}{n}+2\left({n}+{m}\right)-1 $ vertices and $ 6{n}{m}+2{m}+{n}-2 $ edges. Moreover, the result required detailed are available in Tables 3 and 4.
Dimension $ {{\boldsymbol{m}}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{15}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{16}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{18}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}^{{\bf{\infty}}}{\bf{|}} $ |
$ 2 $ | $ 3 $ | $ 2\left({n}-1\right) $ | $ {n}-1 $ | $ 20{n}+7 $ |
Dimension $ {{\boldsymbol{m}}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{15}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{16}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{17}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{18}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}^{{\bf{\infty}}}{\bf{|}} $ |
$ 2 $ | $ 3 $ | $ 2\left({n}-1\right) $ | $ 0 $ | $ {n}-1 $ | $ 20{n}+7 $ |
$ 3 $ | $ 2 $ | $ 2{n} $ | $ 1 $ | $ 3\left({n}-1\right) $ | $ 20{n}+17 $ |
$ 4 $ | $ 2 $ | $ 2{n} $ | $ 3 $ | $ 5\left({n}-1\right) $ | $ 20{n}+27 $ |
$ 5 $ | $ 2 $ | $ 2{n} $ | $ 5 $ | $ 7\left({n}-1\right) $ | $ 20{n}+37 $ |
$ 6 $ | $ 2 $ | $ 2{n} $ | $ 7 $ | $ 9\left({n}-1\right) $ | $ 20{n}+47 $ |
. | . | . | . | . | . |
. | . | . | . | . | . |
. | . | . | . | . | . |
$ {m} $ | $ 2 $ | $ 2{n} $ | $ 2{m}-5 $ | $ 2{m}{n}-2{m}-3{n}+3 $ | $ 20{n}+10{m}-13 $ |
Theorem 2.4. Let $ HCS_{{m}, {n}} $ be the carbon sheet of dimension $ \left({m}, {n}\right) $ and $ {m}, {n}\geq2. $ Then the face index of $ HCS_{{m}, {n}} $ is
$ FI(HCSm,n)={70n+2ifm=236mn−14−2(n−4m)ifm≥3. $
|
Proof. Consider $ HCS_{{m}, {n}} $ be the carbon sheet of dimension $ \left({m}, {n}\right) $ and $ {m}, {n}\geq2. $ Let $ {f}_{i} $ denotes the faces of graph $ HCS_{{m}, {n}} $ having degree $ {i}, $ which is $ d(f_{i}) = \sum_{\alpha\sim f_{i}}{d\left(\alpha\right)} = i, $ and $ |f_{i}| $ denotes the number of faces with degree $ i. $ A graph $ HCS_{{m}, {n}} $ for a particular value of $ {m} = 2 $ contains three types of internal faces $ {f}_{15}, $ $ {f}_{16}, $ $ {f}_{17} $ and $ {f}_{18} $ with one external face $ {f}^{\infty}. $ While for the generalize values of $ {m}\geq3, $ it contain four types of internal faces $ {f}_{15}, $ $ {f}_{16} $ and $ {f}_{17} $ with one external face $ {f}^{\infty} $ in usual manner. For the face index of generalize nanotube, we will divide into two cases on the values of $ {m}. $
Case 1: When $ HCS_{{m}, {n}} $ has one row or $ HCS_{{2}, {n}}. $
A graph $ HCS_{{m}, {n}} $ for a this particular value of $ {m} = 2 $ contains three types of internal faces $ \left|{f}_{15}\right| = 3, $ $ \left|{f}_{16}\right| = 2\left({n}-1\right) $ and $ \left|{f}_{18}\right| = {n}-1 $ with one external face $ {f}^{\infty}. $ For the face index of carbon sheet, details are given in the Table 3. Now the face index $ FI $ of the graph $ NT_{{2}, {n}} $ is given by
$ FI(HCS2,n)=∑α∼f∈F(HCS2,n)d(α)=∑α∼f15∈F(HCS2,n)d(α)+∑α∼f16∈F(HCS2,n)d(α)+∑α∼f18∈F(HCS2,n)d(α)+∑α∼f∞∈F(HCS2,n)d(α)=|f15|(15)+|f16|(16)+|f18|(18)+20n+7.=3(15)+2(n−1)(16)+(n−1)(18)+20n+7.=70n+2. $
|
Case 2: When $ HCS_{{m}, {n}} $ has $ {m}\geq3 $ rows.
A graph $ HCS_{{m}, {n}} $ for generalize values of $ {m}\geq3 $ contains four types of internal faces $ \left|{f}_{15}\right| = 2, $ $ \left|{f}_{16}\right| = 2{n}, $ $ \left|{f}_{17}\right| = 2{m}-5 $ and $ \left|{f}_{18}\right| = 2{m}{n}-2{m}-3{n}+3 $ with one external face $ {f}^{\infty}. $ For the face index of carbon sheet, details are given in the Table 4. Now the face index $ FI $ of the graph $ NT_{{m}, {n}} $ is given by
$ FI(HCSm,n)=∑α∼f∈F(HCSm,n)d(α)=∑α∼f15∈F(HCSm,n)d(α)+∑α∼f16∈F(HCSm,n)d(α)+∑α∼f17∈F(HCSm,n)d(α)+∑α∼f18∈F(HCSm,n)d(α)+∑α∼f∞∈F(HCSm,n)d(α)=|f15|(15)+|f16|(16)+|f17|(17)+|f18|(18)+20n+10m−13.=36mn−2n+8m−14. $
|
Given structure of polyhedron generalized sheet of $ C^{\ast}_{28} $ in the Figure 7, is made by generalizing a $ C^{\ast}_{28} $ polyhedron structure which is shown in the Figure 8. This particular structure of $ C^{\ast}_{28} $ polyhedron are given in [57]. The polyhedron generalized sheet of $ C^{\ast}_{28} $ is as symbolize $ PHS_{{m}, {n}}, $ where $ {n} $ represents the total number of vertical $ C^{\ast}_{28} $ polyhedrons and $ {m} $ denotes the horizontal $ C^{\ast}_{28} $ polyhedrons. It contain total $ 23{n}{m}+3{n}+2{m} $ vertices and $ 33{n}{m}+{n}+{m} $ edges. Moreover, the result required detailed are available in Tables 3 and 5.
$ {{\boldsymbol{m}}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{14}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{15}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{16}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{17}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{18}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{20}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{35}}}{\bf{|}} $ |
$ 1 $ | $ 2{n}+1 $ | $ 2 $ | $ 4{n}-2 $ | $ 0 $ | $ 0 $ | $ 2{n}-1 $ | $ 0 $ |
$ 2 $ | $ 2{n}+2 $ | $ 2 $ | $ 8{n}-2 $ | $ 2 $ | $ 2{n}-2 $ | $ 4{n}-2 $ | $ 2{n}-1 $ |
$ 3 $ | $ 2{n}+3 $ | $ 2 $ | $ 12{n}-2 $ | $ 4 $ | $ 4{n}-4 $ | $ 6{n}-3 $ | $ 4{n}-2 $ |
. | . | . | . | . | . | . | . |
. | . | . | . | . | . | . | . |
. | . | . | . | . | . | . | . |
$ {m} $ | $ 2{n}+{m} $ | $ 2 $ | $ 4{m}{n}-2 $ | $ 2{m}-2 $ | $ 2{m}{n}-2\left({m}+{n}\right)+2 $ | $ 2{m}{n}-{m} $ | $ 2{m}{n}-\left({m}+2{n}\right)+1 $ |
Theorem 2.5. Let $ PHS_{{m}, {n}} $ be the polyhedron generalized sheet of $ C^{\ast}_{28} $ of dimension $ \left({m}, {n}\right) $ and $ {m}, {n}\geq1. $ Then the face index of $ PHS_{{m}, {n}} $ is
$ FI(PHSm,n)=210mn−2(3m+5n). $
|
Proof. Consider $ PHS_{{m}, {n}} $ be the polyhedron generalized sheet of $ C^{\ast}_{28} $ of dimension $ \left({m}, {n}\right) $ and $ {m}, {n}\geq1. $ Let $ {f}_{i} $ denotes the faces of graph $ PHS_{{m}, {n}} $ having degree $ {i}, $ which is $ d(f_{i}) = \sum_{\alpha\sim f_{i}}{d\left(\alpha\right)} = i, $ and $ |f_{i}| $ denotes the number of faces with degree $ i. $ A graph $ PHS_{{m}, {n}} $ for the generalize values of $ {m}, {n}\geq1, $ it contain seven types of internal faces $ {f}_{14}, {f}_{15}, {f}_{16}, {f}_{17}, {f}_{18}, {f}_{20} $ and $ {f}_{35} $ with one external face $ {f}^{\infty} $ in usual manner. For the face index of polyhedron generalized sheet, details are given in the Table 5.
A graph $ PHS_{{m}, {n}} $ for generalize values of $ {m}, {n}\geq1 $ contains seven types of internal faces $ \left|{f}_{14}\right| = 2{n}+{m}, $ $ \left|{f}_{15}\right| = 2, $ $ \left|{f}_{16}\right| = 4{n}{m}-2, $ $ \left|{f}_{17}\right| = 2\left({m}-1\right), $ $ \left|{f}_{18}\right| = 2{n}{m}-2\left({m}+{n}\right)+2, $ $ \left|{f}_{20}\right| = 2{n}{m}-2{m}{n}-{m}, $ and $ \left|{f}_{35}\right| = 2{m}{n}-{m}-2{n}+1 $ with one external face $ {f}^{\infty}. $ Now the face index $ FI $ of the graph $ PHS_{{m}, {n}} $ is given by
$ FI(PHSm,n)=∑α∼f∈F(PHSm,n)d(α)=∑α∼f14∈F(PHSm,n)d(α)+∑α∼f15∈F(PHSm,n)d(α)+∑α∼f16∈F(PHSm,n)d(α)+∑α∼f17∈F(PHSm,n)d(α)+∑α∼f18∈F(PHSm,n)d(α)+∑α∼f20∈F(PHSm,n)d(α)+∑α∼f35∈F(PHSm,n)d(α)+∑α∼f∞∈F(PHSm,n)d(α)=|f14|(14)+|f15|(15)+|f16|(16)+|f17|(17)+|f18|(18)+|f20|(20)+|f35|(35)+37m+68n−35.=210mn−6m−10n. $
|
With the advancement of technology, types of equipment and apparatuses of studying different chemical compounds are evolved. But topological descriptors or indices are still preferable and useful tools to develop numerical science of compounds. Therefore, from time to time new topological indices are introduced to study different chemical compounds deeply. In this study, we discussed a newly developed tool of some silicate type networks and generalized sheets, carbon sheet, polyhedron generalized sheet, with the face index concept. It provides numerical values of these networks based on the information of faces. It also helps to study physicochemical characteristics based on the faces of silicate networks.
M. K. Jamil conceived of the presented idea. K. Dawood developed the theory and performed the computations. M. Azeem verified the analytical methods, R. Luo investigated and supervised the findings of this work. All authors discussed the results and contributed to the final manuscript.
This work was supported by the National Science Foundation of China (11961021 and 11561019), Guangxi Natural Science Foundation (2020GXNSFAA159084), and Hechi University Research Fund for Advanced Talents (2019GCC005).
The authors declare that they have no conflicts of interest.
[1] | Morton NE (1997) Genetic epidemiology. Ann Hum Genet 61: 1-13. |
[2] | Morton NE (1994) Fundamentals of genetic epidemiology. Genet Epidemiol 11: 389-390. |
[3] | Morton NE (1982) Outline of genetic epidemiology. S. Karger AG (Switzerland), 252. |
[4] | Cohen BH (1980) Chronic obstructive pulmonary disease: A challenge in genetic epidemiology. Am J Epidemiol 112: 274-288. |
[5] | Morey M, Fernández-Marmiesse A, Castiñeiras D, et al. (2013) A glimpse into past, present, and future DNA sequencing. Mol Genet Metab 110: 3-24. |
[6] | Matullo G, Gaetano CD, Guarrera S (2013) Next generation sequencing and rare genetic variants: From human population studies to medical genetics. Environ Mol Mutagen 54: 518-532. |
[7] | IJzerman RG, Stehouwer CDA, Boomsma DI (2000) Evidence for genetic factors explaining the birth Weight–Blood pressure relation: Analysis in twins. Hypertension 36: 1008-1012. |
[8] | Ostern R, Fagerheim T, Hjellnes H, et al. (2014) Segregation analysis in families with charcot-marie-tooth disease allows reclassification of putative disease causing mutations. BMC Med Genet 15: 12. |
[9] | Jorde LB (2000) Linkage disequilibrium and the search for complex disease genes. Genome Res 10: 1435-1444. |
[10] | Guo SW (2001) Does higher concordance in monozygotic twins than in dizygotic twins suggest a genetic component?. Hum Hered 51: 121-132. |
[11] | King RC, Mulligan P, Stansfield W (2013) A dictionary of genetics. Oxford University Press, 641. |
[12] | Wong AHC, Gottesman II, Petronis A (2005) Phenotypic differences in genetically identical organisms: The epigenetic perspective. Hum Mol Genet 14: R11-18. |
[13] | Chaganti RSK, Miller DR, Meyers PA, et al. (1979) Cytogenetic evidence of the intrauterine origin of acute leukemia in monozygotic twins. N Engl J Med 300: 1032-1034. |
[14] | Bell JT, Saffery R (2012) The value of twins in epigenetic epidemiology. Int J Epidemiol 41: 140-150. |
[15] | Elston RC (1981) Segregation analysis. In: Harris H and Hirschhorn K, eds. Springer US, 63-120. |
[16] | Jarvik GP (1998) Complex segregation analyses: Uses and limitations. Am J Hum Genet 63: 942-946. |
[17] | Terwilliger JD, Goring HH (2000) Gene mapping in the 20th and 21st centuries: Statistical methods, data analysis, and experimental design. Hum Biol 72: 63-132. |
[18] | Bateson W, Waunders ER, Punnett RC (1909) Experimental studies in the physiology of heredity. Zeitschrift für Induktive Abstammungs- Und Vererbungslehre 2: 17-19. |
[19] | Stevens WL (1939) Tables of the recombination fraction estimated from the product ratio. J Genet 39: 171-180. |
[20] | Tan YD, Fu YX (2007) A new strategy for estimating recombination fractions between dominant markers from an F2 population. Genetics 175: 923-931. |
[21] | Botstein D, White RL, Skolnick M, et al. (1980) Construction of a genetic linkage map in man using restriction fragment length polymorphisms. Am J Hum Genet 32: 314-331. |
[22] |
Stocker AJ, Rusuwa BB, Blacket MJ, et al. (2012) Physical and linkage maps for drosophila serrata, a model species for studies of clinal adaptation and sexual selection. G3 (Bethesda) 2: 287-297. doi: 10.1534/g3.111.001354
![]() |
[23] | Bailey-Wilson JE (2005) Parametric versus nonparametric and two-point versus multipoint: Controversies in gene mapping. In: Anonymous Encyclopedia of Genetics, Genomics, Proteomics and Bioinformatics. John Wiley & Sons, Ltd. |
[24] | Hirschhorn JN, Lohmueller K, Byrne E, et al. (2002) A comprehensive review of genetic association studies. Genet Med 4: 45-61. |
[25] | Cordell HJ, Clayton DG (2005) Genetic association studies. Lancet 366: 1121-1131. |
[26] | McCarthy MI, Abecasis GR, Cardon LR, et al. (2008) Genome-wide association studies for complex traits: Consensus, uncertainty and challenges. Nat Rev Genet 9: 356-369. |
[27] | St George-Hyslop PH, Haines JL, Farrer LA, et al. (1990) Genetic linkage studies suggest that alzheimer's disease is not a single homogeneous disorder. Nature 347: 194-197. |
[28] | Klein RJ, Zeiss C, Chew EY, et al. (2005) Complement factor H polymorphism in age-related macular degeneration. Science 308: 385-389. |
[29] | Welter D, MacArthur J, Morales J, et al. (2013) The NHGRI GWAS catalog, a curated resource of SNP-trait associations. Nucleic Acids Res 42: D1001-1006. |
[30] | Kooperberg C, LeBlanc M, Obenchain V (2010) Risk prediction using genome-wide association studies. Genet Epidemiol 34: 643-652. |
[31] | Gusev A, Bhatia G, Zaitlen N, et al. (2013) Quantifying missing heritability at known GWAS loci. PLoS Genet 9: e1003993. |
[32] | Stranger BE, Stahl EA, Raj T (2011) Progress and promise of genome-wide association studies for human complex trait genetics. Genetics 187: 367-383. |
[33] | Visscher P, Brown M, McCarthy M, et al. (2012) Five years of GWAS discovery. Am J Hum Genet 90: 7-24. |
[34] | Gibson G (2012) Rare and common variants: Twenty arguments. Nat Rev Genet 13: 135-145. |
[35] | Slatkin M (2008) Linkage disequilibrium - understanding the evolutionary past and mapping the medical future. Nat Rev Genet 9: 477-485. |
[36] | Cooper GM, Shendure J (2011) Needles in stacks of needles: Finding disease-causal variants in a wealth of genomic data. Nat Rev Genet 12: 628-640. |
[37] | Manolio TA, Collins FS, Cox NJ, et al. (2009) Finding the missing heritability of complex diseases. Nature 461: 747-753. |
[38] | Frazer KA, Murray SS, Schork NJ, et al. (2009) Human genetic variation and its contribution to complex traits. Nat Rev Genet 10: 241-251. |
[39] | Johnson DS, Mortazavi A, Myers RM, et al. (2007) Genome-wide mapping of in vivo protein-DNA interactions. Science 316: 1497-1502. |
[40] | Shen P, Wang W, Krishnakumar S, et al. (2011) High-quality DNA sequence capture of 524 disease candidate genes. Proc Natl Acad Sci U S A 108: 6549-6554. |
[41] | Service RF (2006) The race for the $1000 genome. Science 311: 1544-1546. |
[42] | Wetterstrand KA, DNA Sequencing Costs: Data from the NHGRI Large-Scale Genome Sequencing Program. 2015. Available from: www.genome.gov/sequencingcosts |
[43] | Broadwith P (2012) Sequencing in the fast lane. Chem World 9: 54-58. |
[44] | Feldman AL, Dogan A, Smith DI, et al. (2010) Massively parallel mate pair DNA library sequencing for translocation discovery: Recurrent t(6;7)(p25.3;q32.3) translocations in ALK-negative anaplastic large cell lymphomas. ASH Annual Meeting Abstracts 116: 633. |
[45] | Green R, Malaspinas A, Krause J, et al. (2008) A complete neandertal mitochondrial genome sequence determined by high-throughput sequencing. Cell 134: 416-426. |
[46] | Durbin RM, Altshuler DL, Durbin RM, et al. (2010) A map of human genome variation from population-scale sequencing. Nature 467: 1061-1073. |
[47] | Peters BA, Kermani BG, Sparks AB, et al. (2012) Accurate whole-genome sequencing and haplotyping from 10 to 20 human cells. Nature 487: 190-195. |
[48] | Butler J, MacCallum I, Kleber M, et al. (2008) ALLPATHS: De novo assembly of whole-genome shotgun microreads. Genome Res 18: 810-820. |
[49] | Furlotte NA, Heckerman D, Lippert C (2014) Quantifying the uncertainty in heritability. J Hum Genet 59: 269-275. |
[50] | Majewski J, Schwartzentruber J, Lalonde E, et al. (2011) What can exome sequencing do for you?. J Med Genet 48: 580-589. |
[51] | Wooderchak-Donahue W, O’Fallon B, Furtado L, et al. (2012) A direct comparison of next generation sequencing enrichment methods using an aortopathy gene panel- clinical diagnostics perspective. BMC Medical Genomics 5: 1-10. |
[52] | Kalender Atak Z, De Keersmaecker K, Gianfelici V, et al. (2012) High accuracy mutation detection in leukemia on a selected panel of cancer genes. PLoS One 7: e38463. |
[53] | Ni T, Wu H, Song S, et al. (2009) Selective gene amplification for high-throughput sequencing. Recent Pat DNA Gene Seq 3: 29-38. |
[54] | Gaugler T, Klei L, Sanders SJ, et al. (2014) Most genetic risk for autism resides with common variation. Nat Genet 46: 881-885. |
[55] | Muona M, Berkovic SF, Dibbens LM, et al. (2015) A recurrent de novo mutation in KCNC1 causes progressive myoclonus epilepsy. Nat Genet 47: 39-46. |
[56] | Xu B, Ionita-Laza I, Roos JL, et al. (2012) De novo gene mutations highlight patterns of genetic and neural complexity in schizophrenia. Nat Genet 44: 1365-1369. |
[57] | Cardinale CJ, Kelsen JR, Baldassano RN, et al. (2013) Impact of exome sequencing in inflammatory bowel disease. World J Gastroenterol 19: 6721-6729. |
[58] | Gilissen C, Arts HH, Hoischen A, et al. (2010) Exome sequencing identifies WDR35 variants involved in sensenbrenner syndrome. Am J Hum Genet 87: 418-423. |
[59] | Boycott KM, Vanstone MR, Bulman DE, et al. (2013) Rare-disease genetics in the era of next-generation sequencing: Discovery to translation. Nat Rev Genet 14: 681-691. |
[60] | Roach JC, Glusman G, Smit AF, et al. (2010) Analysis of genetic inheritance in a family quartet by whole-genome sequencing. Science 328: 636-639. |
[61] | Fernandez-Marmiesse A, Morey M, Pineda M, et al. (2014) Assessment of a targeted resequencing assay as a support tool in the diagnosis of lysosomal storage disorders. Orphanet J Rare Dis 9: 59. |
[62] | Audo I, Bujakowska KM, Leveillard T, et al. (2012) Development and application of a next-generation-sequencing (NGS) approach to detect known and novel gene defects underlying retinal diseases. Orphanet J Rare Dis 7: 8. |
[63] | Mardis ER (2009) New strategies and emerging technologies for massively parallel sequencing: Applications in medical research. Genome Med 1: 40. |
[64] | Wendl MC, Wilson RK (2009) The theory of discovering rare variants via DNA sequencing. BMC Genomics 10: 485. |
[65] | Cooper DN, Krawczak M, Polychronakos C, et al. (2013) Where genotype is not predictive of phenotype: Towards an understanding of the molecular basis of reduced penetrance in human inherited disease. Hum Genet 132: 1077-1130. |
[66] | Venter JC, Adams MD, Myers EW, et al. (2001) The sequence of the human genome. Science 291: 1304-1351. |
[67] | Lander ES, Linton LM, Birren B, et al. (2001) Initial sequencing and analysis of the human genome. Nature 409: 860-921. |
[68] | Durbin R, Altshuler D, Durbin R, et al. (2010) A map of human genome variation from population-scale sequencing. Nature 467: 1061-1073. |
[69] | Panoutsopoulou K, Tachmazidou I, Zeggini E (2013) In search of low-frequency and rare variants affecting complex traits. Hum Mol Genet 22: R16-21. |
[70] | Li J, Schmieder R, Ward RM, et al. (2012) SEQanswers: An open access community for collaboratively decoding genomes. Bioinformatics 28: 1272-1273. |
[71] | Elgar G, Vavouri T (2008) Tuning in to the signals: Noncoding sequence conservation in vertebrate genomes. Trends Genet 24: 344-352. |
[72] | Eisenberger T, Neuhaus C, Khan AO, et al. (2013) Increasing the yield in targeted next-generation sequencing by implicating CNV analysis, non-coding exons and the overall variant load: The example of retinal dystrophies. PLoS One 8: e78496. |
[73] | Ward LD, Kellis M (2011) HaploReg: A resource for exploring chromatin states, conservation, and regulatory motif alterations within sets of genetically linked variants. Nucleic Acids Res 40: D930-934. |
[74] | Boyle AP, Hong EL, Hariharan M, et al. (2012) Annotation of functional variation in personal genomes using RegulomeDB. Genome Res 22: 1790-1797. |
[75] | Kircher M, Witten DM, Jain P, et al. (2014) A general framework for estimating the relative pathogenicity of human genetic variants. Nat Genet 46: 310-315. |
[76] | Vandeweyer G, Van Laer L, Loeys B, et al. (2014) VariantDB: A flexible annotation and filtering portal for next generation sequencing data. Genome Med 6: 74. |
[77] | Ritchie GRS, Dunham I, Zeggini E, et al. (2014) Functional annotation of noncoding sequence variants. Nat Meth 11: 294-296. |
[78] | Wang K, Li M, Hakonarson H (2010) ANNOVAR: Functional annotation of genetic variants from high-throughput sequencing data. Nucleic Acids Research 38: e164-. |
[79] | Henry VJ, Bandrowski AE, Pepin A, et al. (2014) OMICtools: An informative directory for multi-omic data analysis. Database (Oxford) bau069. |
[80] | The ENCODE Project Consortium (2012) An integrated encyclopedia of DNA elements in the human genome. Nature 489: 57-74. |
[81] | The 1000 Genomes Project Consortium (2012) An integrated map of genetic variation from 1,092 human genomes. Nature 491: 56-65. |
[82] |
McEwen JE, Boyer JT, Sun KY, et al. (2014) The ethical, legal, and social implications program of the national human genome research institute: Reflections on an ongoing experiment. Annu Rev Genomics Hum Genet 15: 481-505.
|
[83] |
1. | Yasunari Matsuzaka, Ryu Yashiro, Applications of Deep Learning for Drug Discovery Systems with BigData, 2022, 2, 2673-7426, 603, 10.3390/biomedinformatics2040039 | |
2. | Hao Wang, Guangmin Sun, Kun Zheng, Hui Li, Jie Liu, Yu Bai, Privacy protection generalization with adversarial fusion, 2022, 19, 1551-0018, 7314, 10.3934/mbe.2022345 | |
3. | Yu Li, Hao Liang, Guangmin Sun, Zifeng Yuan, Yuanzhi Zhang, Hongsheng Zhang, A Land Cover Background-Adaptive Framework for Large-Scale Road Extraction, 2022, 14, 2072-4292, 5114, 10.3390/rs14205114 | |
4. | Haiying Yuan, Mengfan Dai, Cheng Shi, Minghao Li, Haihang Li, A generative adversarial neural network with multi-attention feature extraction for fundus lesion segmentation, 2023, 43, 1573-2630, 5079, 10.1007/s10792-023-02911-y | |
5. | Xue Xia, Kun Zhan, Yuming Fang, Wenhui Jiang, Fei Shen, Lesion‐aware network for diabetic retinopathy diagnosis, 2023, 33, 0899-9457, 1914, 10.1002/ima.22933 | |
6. | Tiwalade Modupe Usman, Yakub Kayode Saheed, Adeyemi Abel Ajibesin, Augustine Shey Nsang, 2024, Ens5B-UNet for Improved Microaneurysms Segmentation in Retinal Images, 979-8-3503-5815-5, 1, 10.1109/SEB4SDG60871.2024.10629958 | |
7. | Huma Naz, Neelu Jyothi Ahuja, Rahul Nijhawan, Diabetic retinopathy detection using supervised and unsupervised deep learning: a review study, 2024, 57, 1573-7462, 10.1007/s10462-024-10770-x | |
8. | Joshua E. Mckone, Tryphon Lambrou, Xujiong Ye, James M. Brown, Weakly supervised pre-training for brain tumor segmentation using principal axis measurements of tumor burden, 2024, 6, 2624-9898, 10.3389/fcomp.2024.1386514 | |
9. | Tiwalade Modupe Usman, Adeyemi Abel Ajibesin, Yakub Kayode Saheed, Augustine Shey Nsang, 2023, GAPS-U-NET: Gating Attention And Pixel Shuffling U-Net For Optic Disc Segmentation In Retinal Images, 979-8-3503-5883-4, 1, 10.1109/ICMEAS58693.2023.10429873 |
Dimension | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{12}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{15}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{36}}}{\bf{|}} $ |
$ 1 $ | $ 24 $ | $ 48 $ | $ 7 $ |
$ 2 $ | $ 32 $ | $ 94 $ | $ 14 $ |
$ 3 $ | $ 40 $ | $ 152 $ | $ 23 $ |
$ 4 $ | $ 48 $ | $ 222 $ | $ 34 $ |
$ 5 $ | $ 56 $ | $ 304 $ | $ 47 $ |
$ 6 $ | $ 64 $ | $ 398 $ | $ 62 $ |
$ 7 $ | $ 72 $ | $ 504 $ | $ 79 $ |
$ 8 $ | $ 80 $ | $ 622 $ | $ 98 $ |
. | . | . | . |
. | . | . | . |
. | . | . | . |
$ {n} $ | $ 8{n}+16 $ | $ 6{n}^{2}+28{n}+14 $ | $ {n}^{2}+4{n}+2 $ |
Dimension | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{12}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{14}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{17}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{18}}}{\bf{|}} $ |
$ 1 $ | $ 6 $ | $ 0 $ | $ 0 $ | $ 0 $ |
$ 2 $ | $ 6 $ | $ 12 $ | $ 12 $ | $ 12 $ |
$ 3 $ | $ 6 $ | $ 24 $ | $ 24 $ | $ 60 $ |
$ 4 $ | $ 6 $ | $ 36 $ | $ 36 $ | $ 144 $ |
$ 5 $ | $ 6 $ | $ 48 $ | $ 48 $ | $ 264 $ |
$ 6 $ | $ 6 $ | $ 60 $ | $ 60 $ | $ 420 $ |
$ 7 $ | $ 6 $ | $ 72 $ | $ 72 $ | $ 612 $ |
$ 8 $ | $ 6 $ | $ 84 $ | $ 84 $ | $ 840 $ |
. | . | . | . | . |
. | . | . | . | . |
. | . | . | . | . |
$ {k} $ | $ 6 $ | $ 12({k}-1) $ | $ 12({k}-1) $ | $ 18{k}^{2}-42{k}+24 $ |
Dimension $ {{\boldsymbol{m}}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{15}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{16}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{18}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}^{{\bf{\infty}}}{\bf{|}} $ |
$ 2 $ | $ 3 $ | $ 2\left({n}-1\right) $ | $ {n}-1 $ | $ 20{n}+7 $ |
Dimension $ {{\boldsymbol{m}}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{15}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{16}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{17}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{18}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}^{{\bf{\infty}}}{\bf{|}} $ |
$ 2 $ | $ 3 $ | $ 2\left({n}-1\right) $ | $ 0 $ | $ {n}-1 $ | $ 20{n}+7 $ |
$ 3 $ | $ 2 $ | $ 2{n} $ | $ 1 $ | $ 3\left({n}-1\right) $ | $ 20{n}+17 $ |
$ 4 $ | $ 2 $ | $ 2{n} $ | $ 3 $ | $ 5\left({n}-1\right) $ | $ 20{n}+27 $ |
$ 5 $ | $ 2 $ | $ 2{n} $ | $ 5 $ | $ 7\left({n}-1\right) $ | $ 20{n}+37 $ |
$ 6 $ | $ 2 $ | $ 2{n} $ | $ 7 $ | $ 9\left({n}-1\right) $ | $ 20{n}+47 $ |
. | . | . | . | . | . |
. | . | . | . | . | . |
. | . | . | . | . | . |
$ {m} $ | $ 2 $ | $ 2{n} $ | $ 2{m}-5 $ | $ 2{m}{n}-2{m}-3{n}+3 $ | $ 20{n}+10{m}-13 $ |
$ {{\boldsymbol{m}}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{14}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{15}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{16}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{17}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{18}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{20}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{35}}}{\bf{|}} $ |
$ 1 $ | $ 2{n}+1 $ | $ 2 $ | $ 4{n}-2 $ | $ 0 $ | $ 0 $ | $ 2{n}-1 $ | $ 0 $ |
$ 2 $ | $ 2{n}+2 $ | $ 2 $ | $ 8{n}-2 $ | $ 2 $ | $ 2{n}-2 $ | $ 4{n}-2 $ | $ 2{n}-1 $ |
$ 3 $ | $ 2{n}+3 $ | $ 2 $ | $ 12{n}-2 $ | $ 4 $ | $ 4{n}-4 $ | $ 6{n}-3 $ | $ 4{n}-2 $ |
. | . | . | . | . | . | . | . |
. | . | . | . | . | . | . | . |
. | . | . | . | . | . | . | . |
$ {m} $ | $ 2{n}+{m} $ | $ 2 $ | $ 4{m}{n}-2 $ | $ 2{m}-2 $ | $ 2{m}{n}-2\left({m}+{n}\right)+2 $ | $ 2{m}{n}-{m} $ | $ 2{m}{n}-\left({m}+2{n}\right)+1 $ |
Dimension | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{12}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{15}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{36}}}{\bf{|}} $ |
$ 1 $ | $ 24 $ | $ 48 $ | $ 7 $ |
$ 2 $ | $ 32 $ | $ 94 $ | $ 14 $ |
$ 3 $ | $ 40 $ | $ 152 $ | $ 23 $ |
$ 4 $ | $ 48 $ | $ 222 $ | $ 34 $ |
$ 5 $ | $ 56 $ | $ 304 $ | $ 47 $ |
$ 6 $ | $ 64 $ | $ 398 $ | $ 62 $ |
$ 7 $ | $ 72 $ | $ 504 $ | $ 79 $ |
$ 8 $ | $ 80 $ | $ 622 $ | $ 98 $ |
. | . | . | . |
. | . | . | . |
. | . | . | . |
$ {n} $ | $ 8{n}+16 $ | $ 6{n}^{2}+28{n}+14 $ | $ {n}^{2}+4{n}+2 $ |
Dimension | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{12}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{14}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{17}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{18}}}{\bf{|}} $ |
$ 1 $ | $ 6 $ | $ 0 $ | $ 0 $ | $ 0 $ |
$ 2 $ | $ 6 $ | $ 12 $ | $ 12 $ | $ 12 $ |
$ 3 $ | $ 6 $ | $ 24 $ | $ 24 $ | $ 60 $ |
$ 4 $ | $ 6 $ | $ 36 $ | $ 36 $ | $ 144 $ |
$ 5 $ | $ 6 $ | $ 48 $ | $ 48 $ | $ 264 $ |
$ 6 $ | $ 6 $ | $ 60 $ | $ 60 $ | $ 420 $ |
$ 7 $ | $ 6 $ | $ 72 $ | $ 72 $ | $ 612 $ |
$ 8 $ | $ 6 $ | $ 84 $ | $ 84 $ | $ 840 $ |
. | . | . | . | . |
. | . | . | . | . |
. | . | . | . | . |
$ {k} $ | $ 6 $ | $ 12({k}-1) $ | $ 12({k}-1) $ | $ 18{k}^{2}-42{k}+24 $ |
Dimension $ {{\boldsymbol{m}}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{15}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{16}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{18}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}^{{\bf{\infty}}}{\bf{|}} $ |
$ 2 $ | $ 3 $ | $ 2\left({n}-1\right) $ | $ {n}-1 $ | $ 20{n}+7 $ |
Dimension $ {{\boldsymbol{m}}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{15}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{16}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{17}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{18}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}^{{\bf{\infty}}}{\bf{|}} $ |
$ 2 $ | $ 3 $ | $ 2\left({n}-1\right) $ | $ 0 $ | $ {n}-1 $ | $ 20{n}+7 $ |
$ 3 $ | $ 2 $ | $ 2{n} $ | $ 1 $ | $ 3\left({n}-1\right) $ | $ 20{n}+17 $ |
$ 4 $ | $ 2 $ | $ 2{n} $ | $ 3 $ | $ 5\left({n}-1\right) $ | $ 20{n}+27 $ |
$ 5 $ | $ 2 $ | $ 2{n} $ | $ 5 $ | $ 7\left({n}-1\right) $ | $ 20{n}+37 $ |
$ 6 $ | $ 2 $ | $ 2{n} $ | $ 7 $ | $ 9\left({n}-1\right) $ | $ 20{n}+47 $ |
. | . | . | . | . | . |
. | . | . | . | . | . |
. | . | . | . | . | . |
$ {m} $ | $ 2 $ | $ 2{n} $ | $ 2{m}-5 $ | $ 2{m}{n}-2{m}-3{n}+3 $ | $ 20{n}+10{m}-13 $ |
$ {{\boldsymbol{m}}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{14}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{15}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{16}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{17}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{18}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{20}}}{\bf{|}} $ | $ {\bf{|}}{\boldsymbol{f}}_{{\bf{35}}}{\bf{|}} $ |
$ 1 $ | $ 2{n}+1 $ | $ 2 $ | $ 4{n}-2 $ | $ 0 $ | $ 0 $ | $ 2{n}-1 $ | $ 0 $ |
$ 2 $ | $ 2{n}+2 $ | $ 2 $ | $ 8{n}-2 $ | $ 2 $ | $ 2{n}-2 $ | $ 4{n}-2 $ | $ 2{n}-1 $ |
$ 3 $ | $ 2{n}+3 $ | $ 2 $ | $ 12{n}-2 $ | $ 4 $ | $ 4{n}-4 $ | $ 6{n}-3 $ | $ 4{n}-2 $ |
. | . | . | . | . | . | . | . |
. | . | . | . | . | . | . | . |
. | . | . | . | . | . | . | . |
$ {m} $ | $ 2{n}+{m} $ | $ 2 $ | $ 4{m}{n}-2 $ | $ 2{m}-2 $ | $ 2{m}{n}-2\left({m}+{n}\right)+2 $ | $ 2{m}{n}-{m} $ | $ 2{m}{n}-\left({m}+2{n}\right)+1 $ |