A feature fusion-based attention graph convolutional network for 3D classification and segmentation

1.   Introduction

The motivation of this study arises from the molecular descriptors and their conversion into number structures. It enables Mathematics to play its part in these studies. This work is useful in Chemistry, pharmacy, and environmental protection. The {topological indices} which are molecular descriptors that remain invariants in the molecular structure created a bridge between chemical structures through the characterization of chemical diagrams with graph theory and combinatorics. It also attracts algebraists and other researchers to correlate algebraic structures to enhance the parameter of mentioned applications.

The molecular graph can be considered as the structural formula for a chemical compound that can be seen as a graph structure. A molecular graph can be seen as a colored graph where atoms are considered to be vertices of the compound and edges are considered as atomic bonds. We consider a topological index for a graph of a molecular structure as a real digit that is assigned in a way that it represents the topological structure of the graph and remains constant up to the graph automorphism. The indices associated with a molecular graph structure infers several applications in chemistry. It also infers the study of nanotube structures that can be seen in ^{[10,12,22,24]}.

The characteristic of a topological index is the association of various types of topological indices to a graph. Topological indices are mainly associated with the degrees, distances and eccentricities of a graph. In particular, index related to Randi connectivity, harmonic indices, connectivity of atomic bond, Zagreb indices and geometric arithmetic indices are based on the degrees ^[1,2,5,28].

Some of the examples of distance-based topological indices are the Hosaya index, Wiener index, Estrada index ^[36,37]. The geometric-arithmetic eccentric index ^[20], Zagreb eccentric index ^[19,35], connectivity of atomic bond eccentric index ^[15] and index related to eccentric harmonic ^[16,17] are eccentric based indices.

From the application point of view, the $ABC$ index gives an association between branched and linear alkanes stability. Also, it is used to compute the strain energy for Cycloalkanes ^[18,33]. The $GA$ index is used as a tool to correlate certain Physico-chemical characteristics which is proven to be more effective for predictive power in comparison with the index of Randic connectivity ^[11,32,34]. First, as well as second Zagreb indices are used for the computation of total energy of $p$-electron in a molecule ^[31]. The topological indices which are based on degrees are generally used to analyze the chemical properties of various molecular diagrams.

Motivated by the above works the authors study eccentric-based topological indices for a class of graphs that are being used to analyze the molecule's structure of a compound for the assessment of pharmacological, Physicochemical, and toxicological characteristics. Further details can be seen in ^[14,25]. The QSAR, that is, quantitative structure and activity relationship is used for such analysis ^[21].

For the structural study, ring structures are associated with graphs in several ways, amongst those we shall consider zero-divisor graphs. Initially, these graphs over commutative rings were constructed by I. Beck ^[13] in 1988 and discussed the coloring problem on these graphs.

Classical and logical algebraic structures linked with Graph theory have been studied intensively in previous years for associated invariants and applications in various fields. For example, it is interesting to explore ring $\mathcal{R}$ so that the graph $\Omega(\mathcal{R})$ becomes isomorphic with the given graph $\Gamma.$ By taking a commutative ring having 14 elements and assuming them as vertices, Redmond ^[29] constructed all possible zero-divisor graphs of this structure. Further, an algorithm is provided to compute rings (up to isomorphism) that produce the graph on zero-divisors of a ring having a fixed given number of vertices. Some of the recent studies on algebraic combinatorics with application in other sciences are available in ^[23,26,27]. Further, applications and the relation between the algebraic structures and chemical graphs can be seen in ^[3,14]. The recent works due to Asir et al. ^[8], Selvakumar et al. ^[30] and Asir et al. ^[9] provided formulas for calculating the Wiener index of zero-divisor graphs of $\mathbb{Z}_n.$ A detailed work on graphs associated with rings structure is given in ^[6].

Consider a class of commutative rings $\mathbb{Z}_{p^n}$ where $p$ is a prime number and consider $\Omega(\mathbb{Z}_{p^n}),$ the corresponding zero-divisor graph. In this text, we provide a method of calculation of eccentric topological indices of zero-divisor graphs $\Omega(\mathbb{Z}_{p^n})$ for a fixed positive integer $n$ and any prime number $p.$

In particular, we found first and third Zagreb eccentric indices, geometrico arithmetic eccentric index, atomic bonding connectivity eccentric index and eccentric harmonic indices of fourth types for zero-divisor graphs associated with the rings $\mathbb{Z}_{p^n}.$

2.   Related definitions and some notations

In this whole text $\Gamma$ will denote a general connected graph with set of vertices $V(\Gamma)$ and edges set $E(\Gamma)$. For any $v\in V(\Gamma),$ the degree $d(v)$ denotes the number of edges connected to the vertex $v.$ The numbers $\Delta(\Gamma)$ and $\delta(\Gamma)$ are maximum and minimum degree of a vertex in a graph $\Gamma.$ The number of edges between the shortest path of any two given vertices $v_1$ and $v_2$ will be denoted by $d(v_1, v_2).$ For a vertex $v\in V(\Gamma),$ we define the eccentricity of $v$ as:

The $T(\Gamma)$ is a topological invariant of the eccentricity for the vertices of graph $\Gamma$.

where $\phi(\varrho_u, \varrho_v) = \phi(\varrho_v, \varrho_u)$ gives a real valued function between two eccentricities $\varrho_u$ and $\varrho_v.$

● If $\phi(\varrho_u, \varrho_v) = (\varrho_u + \varrho_v)^\beta$ for $\beta\in\mathbb{\mathcal{R}}\setminus{0},$ then we say that $\phi(\Gamma)$ is the first Zagreb eccentric index if $\beta = 1$ ^[19,35].

● If $\phi(\varrho_u, \varrho_v) = (\varrho_u \times \varrho_v)^\alpha$ for $\alpha\in\mathbb{\mathcal{R}}\setminus{0},$ then we say that $\phi(\Gamma)$ is the third Zagreb eccentric index if $\alpha = 1$ ^[19,35].

● If $\phi(\varrho_u, \varrho_v) = \frac{2\, \sqrt{\varrho_u \times\varrho_v}}{\varrho_u+\varrho_v}$, it gives the geometric arithmetic eccentric index, $A_4(\Gamma)$ ^[20].

● If $\phi(\varrho_u, \varrho_v) = \sqrt{\frac{\varrho_u+\varrho_v-2}{\varrho_u \times \varrho_v}}$, we obtain the atom-bond connectivity eccentric index $ABC_5(\Gamma)$ ^[15].

● If $\phi(\varrho_u, \varrho_v) = \frac{2}{\varrho_u+\varrho_v}$, then we obtained eccentric harmonic index, $H_4(\Gamma),$ called fourth eccentric harmonic index ^[16,17].

We consider a ring $\mathcal{{R}}$ which is commutative with unity. A zero-divisor is a non-zero element $z\in \mathcal{R}$ if there exists another element $x\in \mathcal{R}, \; x\neq 0$ and we have $z.x = 0.$ Similarly, a unit is an element $a\in \mathcal{R}, \; a\neq 0$ if there exists another element $b\in \mathcal{R}, \; b\neq 0$ and we have $a.b = 1.$ The set $Z(\mathcal{R})$ will denotes the collection of all zero-divisors elements in ring $\mathcal{R}$. If $\mathcal{R}$ is finite and commutative, then it is easy to associate $\mathcal{R}$ with $\Omega_\mathcal{R},$ the zero-divisor graph of $\mathcal{R}$ in a way that $V(\Omega_\mathcal{R}) = Z(\mathcal{R})$ becomes the set of vertices of $\Omega_\mathcal{R}$ and the set $E(\Omega_\mathcal{R})$ will represents the edge set of a zero-divisor graph $\Omega_\mathcal{R}.$ Clearly, any $(x_1, x_2)\in E(\Omega_\mathcal{R})$ if $x_1, x_2\in V(\Omega_\mathcal{R})$ and $x_1.x_2 = 0.$ it is proved by Anderson and Livingston ^[7] that the graph $\Omega_\mathcal{R}$ remains a connected graph irrespective of any commutative ring $\mathcal{R}$ ^[4,14].

3.   Main results

In this text, the under consideration rings are of the form $\mathcal{R} = \mathbb{Z}_m$ for a fix positive integer $m.$ An element $x\in \mathbb{Z}_m\setminus \{0\}$ is a zero-divisor if and only if $gcd(x, m) > 1$ and an element $a\in \mathbb{Z}_m\setminus \{0\}$ is a unit if and only if $gcd(a, m) = 1.$ Therefore any non-zero element in the ring $\mathbb{Z}_m$ is either a zero-divisor or a unit.

Let $p$ be any prime number and $n$ be a a fix positive integer. We consider the finite commutative rings of the form $\mathcal{R} = \mathbb{Z}_{p^n}.$ According to above construction, an element $a\in \mathbb{Z}_{p^n}, \; a\neq 0$ is in fact a zero-divisor if and only if $p$ divides $a$. We partition the set of zero-divisors $Z(\mathbb{Z}_{p^n})$ into the sets, $\Lambda_i = \{u.p^i:\; u\hbox{ is a unit in }\mathbb{Z}_{p^n}\}\subseteq \mathbb{Z}_{p^n}$ that in fact contains elements which are a multiple of $p^i$ but not of $p^{i+1}$. Clearly, the set $Z(\mathbb{Z}_{p^n}) = \bigsqcup\limits_{i = 1}^{n-1}\Lambda_i$ and $|\Lambda_i| = p^{n-i}-p^{n-i-1}$ for each $i = 1, 2, \ldots, n-1$ and therefore $|Z(\mathbb{Z}_{p^n})| = \sum\limits_{i = 1}^{n-1}|\Lambda_i| = p^{n-1}-1.$

We denote associated zero-divisor graph to the ring $\mathcal{R} = \mathbb{Z}_{p^n}$ by $\Omega_\mathcal{R} = \Omega_{\mathbb{Z}_{p^n}}$ with set of vertices $V(\Omega_\mathcal{R}) = Z(\mathbb{Z}_{p^n}).$ Since we consider a zero-divisor to be a non-zero element, so $0\not\in V(\Omega_\mathcal{R}).$

Let us denote $d_{\Lambda_i}(x).$ The degree of a vertex $x\in \Lambda_i$ is obtained in the following result.

Proposition 3.1. For a zero-divisor graph $\Omega_\mathcal{R}$ associated to the ring $\mathcal{R} = \mathbb{Z}_{p^n},$ the degree of vertex $x$ is,

Proof. For any vertex $x\in \Lambda_i$, we have $x.y = 0$ if and only if $y\in \Lambda_j$ for $j\ge n-i$. For $1 \le i \le \lceil\frac{n}{2}\rceil-1$ we get $d_{\Lambda_i}(x) = |\bigsqcup\limits_{j = n-i}^{n-1}\Lambda_j| = \sum\limits_{j = n-i}^{n-1}|\Lambda_j| = p^i-1.$ For $\lceil\frac{n}{2}\rceil \le i \le n-1$ we get $d_{\Lambda_i}(x) = |\bigsqcup\limits_{j = n-i}^{n-1}\Lambda_j-\{x\}| = \sum\limits_{j = n-i}^{n-1}|\Lambda_j|-1 = p^i-1-1 = p^i-2.$

By using the hand shaking lemma and after simplification, we obtained the size of the $\Omega_\mathcal{R}$ in the following theorem.

Proposition 3.2. For a prime number $p$ and a fix positive integer $n\ge 2,$ the size of $\Omega_\mathcal{R}$ is,

Theorem 3.3. Consider a prime number $p$ and a fixed integer $n\ge 2.$ Then the vertex eccentricity of the associated graph $\Omega_\mathcal{R}$ with the ring $\mathcal{R} = \mathbb{Z}_{p^n}$ is either $1$ or $2$.

Proof. Let $d(\Lambda_i, \Lambda_j)$ be the distance between sets $\Lambda_i$ and $\Lambda_j$. For any vertex $\varrho_u^i\in \Lambda_i$ and $\varrho_u^j\in \Lambda_j$, then $d(\varrho_u^i, \varrho_v^j) = 1$ for $i = j$ and $1\le i\le \lceil\frac{n}{2}\rceil-1.$ Also, $d(\varrho_u^i, \varrho_v^j) = 2$ for $i = j$ and $\lceil\frac{n}{2}\rceil\le i\le n-1.$ Now, $d(\varrho_u^i, \varrho_v^j) = 2$ for $i\ne j$ & $1\le i, j\le \lceil\frac{n}{2}\rceil-1$ and $d(\varrho_u^i, \varrho_v^j) = 1$ for $i\ne j$ & $\lceil\frac{n}{2}\rceil\le i, j\le n-1.$ Also, $d(\varrho_u^1, \varrho_v^j) = 2$ if $2\le j\le n-2$ and $d(\varrho_u^1, \varrho_v^j) = 1$ if $j = n-1.$ Therefore, the vertex eccentricity of the graph $\Omega_\mathcal{R}$ is $1$ or at most $2$. It also gives that the diameter, $diam(\Omega_\mathcal{R}) = 2$.

Lemma 3.4. For any prime number $p$ and the graph $\Omega_\mathcal{R}$ associated with the ring $\mathcal{R} = \mathbb{Z}_{p^n}$ we have the followings. For the even integers $n,$

For the odd integers $n,$

Proof. The graph $\Omega_\mathcal{R}$ contains $p^{n-1}-1$ vertices and $\frac{1}{2}\{p^{n-1}(np-n-P)-p^{n-\lceil\frac{n}{2}\rceil}+2$ edges except for $n = p = 2.$ Partitioning the edges of $\Omega_\mathcal{R}$ into different sets we obtained the vertex eccentricity.

It is clear that the set $\mathcal{E}_{r, s}$ keeps those edges that are incident to vertices having eccentricities $r$ and $s.$ By using Proposition 3.1 and Theorem 3.3, we get $|\mathcal{E}_{1, 1}| = \frac{p^{n-1}-3p^\frac{n}{2}+2}{2}$ for $n$ even & $|\mathcal{E}_{1, 1}| = \frac{p^{n-1}-3p^\frac{n-1}{2}+2}{2}$ for $n$ odd, $|\mathcal{E}_{1, 2}| = \frac{p^{n-1}(pn-n-2p)}{2}+p^\frac{n}{2}$ for $n$ even & $|\mathcal{E}_{1, 2}| = \frac{p^{n-1}(np-p-n-1)}{2}+p^\frac{n-1}{2}$ for $n$ odd and $E(\Omega_\mathcal{R}) = \mathcal{E}_{1, 1}\cup \mathcal{E}_{1, 2}$.

Then for even integers $n,$

For odd integers $n,$

The given theorem computes the edge-based values for the eccentric based topological indices of graphs $\Omega_\mathcal{R}.$

Theorem 3.5. For a prime number $p\ne 3,$ the zero-divisor graph $\Omega_\mathcal{R}$ for $\mathcal{R} = \mathbb{Z}_{p^n}.$ The first Zagreb eccentric index of the graph $\Omega_\mathcal{R}$ is;

the third Zagreb eccentric index is;

the geometric-arithmetic eccentric index is;

the atom-bond connectivity eccentric index is;

the eccentric harmonic index of type four is as below:

Proof. For the first Zagreb eccentric indices $M_1^{*}(\Omega_\mathcal{R})$ of graph $\Omega_\mathcal{R}$ we have $\phi(\varrho_u, \varrho_v) = \varrho_u + \varrho_v$. Therefore, $\phi(1, 1) = 2$ and $\phi(1, 2) = 3.$ Thus by Lemma 3.4,

For even integers $n,$

For odd integers $n,$

For the third Zagreb eccentric indices $M_3^{*}(\Omega_\mathcal{R})$ of $\Omega_\mathcal{R}$ we get $\phi(\varrho_u, \varrho_v) = \varrho_u \times \varrho_v$. Therefore, $\phi(1, 1) = 1$ and $\phi(1, 2) = 2$ So by Lemma 3.4,

For $n$ even

For $n$ odd

For the geometric arithmetic eccentric index $GA_4(\Omega_\mathcal{R})$ of the graph $\Omega_\mathcal{R}$, we obtained,

Therefore, $\phi(1, 1) = 1$ and $\phi(1, 2) = \frac{2\, \sqrt{2}}{3}.$

So, for even integers $n,$

For odd integers $n,$

For the atom-bond connectivity eccentric index $ABC_5(\Omega_\mathcal{R})$ of $\Omega_\mathcal{R}$, we obtain $\phi(\varrho_u, \varrho_v) = \sqrt{\frac{\varrho_u+\varrho_v-2}{\varrho_u \times \varrho_v}}$. Thus $\phi(1, 1) = 0$ and $\phi(1, 2) = \frac{1}{\sqrt{2}}$ Therefore,

For $n$ even

For $n$ odd

For eccentric harmonic index of type four $H_4(\Omega_\mathcal{R})$ of the graph $\Omega_\mathcal{R}$, we obtained, $\phi(\varrho_u, \varrho_v) = \frac{2}{\varrho_u+\varrho_v}$, hence, $\phi(1, 1) = 1$ and $\phi(1, 2) = \frac{2}{3}.$

Thus, for even integers $n,$

For odd integers $n,$

4.   Conclusions

We have computed Zagreb eccentric index for the first type, the Zagreb eccentric index for the third type, geometrico-arithmetic eccentric index, atomic bonding connectivity eccentric indices and the eccentric harmonic indices of the fourth type of the graphs that are related to the ring $\mathbb{Z}_{p^n}.$ Our work can be used to study different physical and chemical structures such as carbohydrate, silicon structure, polymer, coating, paint constituent and for various computer network problems.

Conflict of interest

The authors declare no conflict of interest.