Research article Special Issues

Maximum degree and minimum degree spectral radii of some graph operations


  • New results relating to the maximum and minimum degree spectral radii of generalized splitting and shadow graphs have been constructed on the basis of any regular graph, referred as base graph. In particular, we establish the relations of extreme degree spectral radii of generalized splitting and shadow graphs of any regular graph.

    Citation: Xiujun Zhang, Ahmad Bilal, M. Mobeen Munir, Hafiz Mutte ur Rehman. Maximum degree and minimum degree spectral radii of some graph operations[J]. Mathematical Biosciences and Engineering, 2022, 19(10): 10108-10121. doi: 10.3934/mbe.2022473

    Related Papers:

    [1] Edil D. Molina, Paul Bosch, José M. Sigarreta, Eva Tourís . On the variable inverse sum deg index. Mathematical Biosciences and Engineering, 2023, 20(5): 8800-8813. doi: 10.3934/mbe.2023387
    [2] V. R. Kulli, J. A. Méndez-Bermúdez, José M. Rodríguez, José M. Sigarreta . Revan Sombor indices: Analytical and statistical study. Mathematical Biosciences and Engineering, 2023, 20(2): 1801-1819. doi: 10.3934/mbe.2023082
    [3] Xinlin Liu, Viktor Krylov, Su Jun, Natalya Volkova, Anatoliy Sachenko, Galina Shcherbakova, Jacek Woloszyn . Segmentation and identification of spectral and statistical textures for computer medical diagnostics in dermatology. Mathematical Biosciences and Engineering, 2022, 19(7): 6923-6939. doi: 10.3934/mbe.2022326
    [4] A. Newton Licciardi Jr., L.H.A. Monteiro . A network model of social contacts with small-world and scale-free features, tunable connectivity, and geographic restrictions. Mathematical Biosciences and Engineering, 2024, 21(4): 4801-4813. doi: 10.3934/mbe.2024211
    [5] Qingqun Huang, Muhammad Labba, Muhammad Azeem, Muhammad Kamran Jamil, Ricai Luo . Tetrahedral sheets of clay minerals and their edge valency-based entropy measures. Mathematical Biosciences and Engineering, 2023, 20(5): 8068-8084. doi: 10.3934/mbe.2023350
    [6] Wenjuan Guo, Ming Ye, Xining Li, Anke Meyer-Baese, Qimin Zhang . A theta-scheme approximation of basic reproduction number for an age-structured epidemic system in a finite horizon. Mathematical Biosciences and Engineering, 2019, 16(5): 4107-4121. doi: 10.3934/mbe.2019204
    [7] J. A. Méndez-Bermúdez, José M. Rodríguez, José L. Sánchez, José M. Sigarreta . Analytical and computational properties of the variable symmetric division deg index. Mathematical Biosciences and Engineering, 2022, 19(9): 8908-8922. doi: 10.3934/mbe.2022413
    [8] Jair Castro, Ludwin A. Basilio, Gerardo Reyna, Omar Rosario . The differential on operator $ {{\mathcal{S}}({\Gamma})} $. Mathematical Biosciences and Engineering, 2023, 20(7): 11568-11584. doi: 10.3934/mbe.2023513
    [9] Cheng-Peng Li, Cheng Zhonglin, Mobeen Munir, Kalsoom Yasmin, Jia-bao Liu . M-polynomials and topological indices of linear chains of benzene, napthalene and anthracene. Mathematical Biosciences and Engineering, 2020, 17(3): 2384-2398. doi: 10.3934/mbe.2020127
    [10] Muhammad Akram, Saba Siddique, Majed G. Alharbi . Clustering algorithm with strength of connectedness for $ m $-polar fuzzy network models. Mathematical Biosciences and Engineering, 2022, 19(1): 420-455. doi: 10.3934/mbe.2022021
  • New results relating to the maximum and minimum degree spectral radii of generalized splitting and shadow graphs have been constructed on the basis of any regular graph, referred as base graph. In particular, we establish the relations of extreme degree spectral radii of generalized splitting and shadow graphs of any regular graph.



    Let G be graph having vertex set V=V(G) and edge set E=E(G), then the adjacency matrix associated to the graph G, is

    [A(G)]={1        if   vi   is adjacent to  vj,0                  otherwise.

    Sum of absolute values of the eigenvalues associated to A(G) is known as energy of the graph G denoted by E(G) and the Largest eigenvalue of A(G) is called Spectral radius of the graph G and it is denoted by (G). For details and references relating to Spectral radii, follow [2,3,4]. The motivation of E(G) was initiated by Gutman in 1978 [1] but the idea could not get attention until 2000. Since 2003, rapid development in technology and computer awoke significant interests in these areas. The problem of determining extreme values of spectral radius has been extensively investigated, [5]. Partial solutions to these problems can be traced in [6,7,8,9,10,11]. Fiedler and Nikiforov [6] gave tight sufficient conditions for the existence of Hamilton paths and cycles in terms of the spectral radius of graphs or the complement of graphs. Lu et al. [7] studied sufficient conditions for Hamilton paths in connected graphs and Hamilton cycles in bipartite graphs in terms of the spectral radius of a graph. Some other spectral conditions for Hamilton paths and cycles in graphs have been given in [12,13,14,15,16].

    Horn et al.[17] and Gatmacher [18] used matrix analysis to relate it with graph energies. Balkrishnan, in [19], computed the sharp bounds for energy of a k-regular graph and proved that for n3, there always exists two equi-energetic graphs having order 4n which are not co-spectral. Bapat et al. proved that E(G) can not be an odd integer, [20] whereas Pirzada et al. [21] proved that it can not be square root of an odd integer. Jones [22] discussed E(G) of simple graphs relating it with closuring and algebraic connectivity. Different kinds of matrices energies associated to a graph are presented by Meenakshi et al. [23]. Nikiforov [24] obtained various results related to bounds of energies. Samir et al. [25] constructed 1-splitting and 1-shadow graph of any simple connected graph and proved that adjacency energies of these newly constructed graphs is constant multiple of the energies of the original graph. Samir et al. [26] then generalized the idea of 1-splitting and 2-shadow graph to arbitrary s-splitting and s-shadow graph where s>0 and obtained similar kind of general results for adjacency energies. Liu et al. [27] discussed distance and adjacency energies of multi-level wheel networks. Chu et al. [19] computed Laplacian and signless Laplacian spectra and energies of multi-step wheels.

    In 2015, Liu et al. [29] discussed asymptotic Laplacian energy like invariants of lattices. In 2016, Hosamani et al. [30] presented degree sum energy of a graph and obtained some lower bounds for this energy. In 2018, Basavanagoud et al. [31] computed the characteristic polynomial of the degree square sum matrix of graphs obtained by some graph operations as well as some bounds for spectral radius for square sum eigenvalue and degree square sum energy of graphs. In 2018, Rad et al. [32] presented Zagreb energy and related Estrada index of various graphs. In 2019, Gutman et al. [33] discussed graph energy and its applications, featuring about hundred kinds of graph energies and applications in diverse areas. For further details and basic ideas of graph energies, we refer [20,21,34,35,36]. Interconnection and various applications of graph energy in chemistry of unsaturated hydrocarbons can be traced in [37,38,39]. Applications of different graph energies in crystallography can be found in [40,41], theory of macro molecules in [42,43], protein sequences in [44,45,46], biology in [47], applied network analysis in [47,48,49,50,51,52], problems of air transportation in [48], satellite communications in [50] and constructions of spacecrafts in [52].

    In the present article we produce new results about maximum degree spectral radii and minimum degree spectral radii of m-splitting and m-shadow graphs. In fact we relate these spectral radii of new graph operations with spectral radii of original graphs. The article is organized as follows. Section 2 gives basic definitions and terminologies to lay foundations of our results. In Section 3, we derive maximum degree spectral radii and minimum degree spectral radii of generalized splitting graph constructed on any basic graph. In Section 4 we proceed to find similar results but for generalized shadow graph of the given regular graph.

    In this part we outline main ideas and preliminary facts, for details see [53,54]. The matrix M(G) is the maximum degree matrix of the graph G defined in [53] as

    M(G)={max(di,dj),ifviandvjareadjacent,0,elsewhere.}

    Here di and dj are the degrees of vertices vi and vj respectively. Eigenvalues of maximum degree matrix of the graph G are denoted as η1,η2,ηn. Maximum degree spectral radius is defined as

    M(G)=nmaxi=1|ηi|,

    where η1,η2,...,ηn are the eigenvalues of maximum degree matrix. The matrix MI(G) is called the minimum degree matrix of the graph G is defined in [54] as

    MI(G)={min(di,dj),ifviandvjareadjacent,0,elsewhere.}

    Minimum degree spectral radius is defined as

    MI(G)=nmaxi=1|ηi|,

    where η1,η2,...,ηn are the eigenvalues of minimum degree matrix. If we add a new vertex v to each vertex v of the graph G, v is connected to every vertex that is adjacent to v in G then we obtain the splitting graph (spl1(G)). Take two copies G and G of the graph G, then (sh2(G)) is constructed if we join each vertex in G to the neighbors of the corresponding vertices in G. Let UϵRm×n, VϵRp×q the tensor product (Kronecker product), UV is defined as the matrix.

    UV=(a11V...a1nV............an1V...annV)

    Proposition 1.1. Let UϵMm, VϵMn and α be an eigenvalue of U and η be an eigenvalue of V, then αη is an eigenvalue of UV [25]. Now we move towards the main results.

    In this part we relate maximum degree spectral radius and minimum degree spectral radius of generalized splitting graph with original graph G. Here again we emphasis that G is any regular graph.

    Theorem 1. Let G be any n-regular graph and M(Splm(G)) is themaximum degree spectral radius of m-splitting graph G, then

    M(Splm(G))=M(G)((m+1)(1+1+4m)2).

    Proof. Maximum degree matrix is given by M(G) where

    M(G)={max(di,dj),ifviandvjareadjacent,0,elsewhere.}

    M(Splm(G)) can be written in block matrix form as

    M(Splm(G))={(m+1)Mifi=1,j1andj=1,i1,0elsewhere.}
    =M{(m+1)ifi=1,j1andj=1,i1,0elsewhere.}

    Let

    A={(m+1)ifi=1,j1andj=1,i1,0elsewhere.}m+1

    Now we compute the eigenvalues of A Since matrix A is of rank two, so A has two non-zero eigenvalues, say α1 and α2. Obviously,

    α1+α2=tr(A)=m+1. (3.1)

    Considering

    A2={(m+1)3ifi=1andj=1,(m+1)2elsewhere.}m+1

    Then

    α21+α22=tr(A2)=(m+1)3+m((m+1)2). (3.2)

    Solving Eqs (3.1) and (3.2), we have

    α1=(m+1)(1+1+4m)2.

    and

    α2=(m+1)(11+4m)2.

    So we have,

    specA=(0(m+1)(1+1+4m)2(m+1)(11+4m)2m111) (3.3)

    Using Eq (3.3) we have

    specA=(0(m+1)(1+1+4m)2(m+1)(11+4m)2m111).

    Since M(Splm(G))=M(G)A then by Proposition 1.1, we have

    M(Splm(G))=Maxni=1|(specA))ηi|
    M(Splm(G))=Maxni=1|ηi|[(m+1)(1+1+4m)2]
    M(Splm(G))=M(G)((m+1)(1+1+4m)2).

    In the following corollaries, we obtain the maximum degree spectral radii of splitting graphs of Cn, Kn, Cn,n and crown graph.

    Corollary 2. If n3 and G is a Cn graph, where Cn is cycle graph on n vertices, then

    M(Splm(Cn))=4((m+1)(1+1+4m)2).

    Proof. If G is a cycle graph Cn(n3), then M(Cn)=4. Since cycle graph is 2-regular graph so using Theorem 1 we get the required result.

    Corollary 3. If G is a Kn graph, where Kn is complete graph on n vertices, then

    M(Splm(Kn))=(n1)2((m+1)(1+1+4m)2).

    Proof. If G is a complete graph on n vertices, then M(Kn)=(n1)2. Since complete graph is n1-regular graph so using Theorem 1 we get the required result.

    Corollary 4. If G is a complete bipartite graph Kn,n, then

    M(Splm(Kn,n))=(n)2((m+1)(1+1+4m)2).

    Proof. If G is a complete bipartite graph Kn,n, then M(Kn,n)=(n)2. Using Theorem 1 we get the required result.

    Corollary 5. If G is a crown graph on 2n vertices, then

    M(Splm(G))=(n1)2((m+1)(1+1+4m)2).

    Proof. If G is a crown graph on 2n vertices, then M(G)=(n1)2. Using Theorem 1 we get the required result.

    Theorem 6. Let G be any n-regular graph and MI((Splm(G)) is theminimum degree spectral radius of m-splitting graph G, then

    MI(Splm(G))=MI(G)(m+1+m2+6m+12).

    Proof. Minimum degree matrix is given by

    MI(G)={min(di,dj),ifviandvjareadjacent,0,elsewhere.}

    MI(Splm(G)) can be written in block matrix form as follows:

    MI(Splm(G))={(m+1)MIifi=1andj=1MIifi=1,j2andj=1,i2,0elsewhere.}
    =MI{(m+1)ifi=1andj=11ifi=1,j2andj=1,i2,0elsewhere.}

    Let

    A={(m+1)ifi=1andj=11ifi=1,j2andj=1,i2,0elsewhere.}m+1

    Now we compute the eigenvalues of A Since matrix A is of rank two, so A has two non-zero eigenvalues, say α1 and α2. Obviously,

    α1+α2=tr(A)=m+1. (3.4)

    Considering

    A2={(m+1)2+mifi=1andj=1,m+1ifi=1,j2andj=1,i2,1elsewhere.}m+1

    Then

    α21+α22=tr(A2)=(m+1)2+2m. (3.5)

    Solving Eqs (3.4) and (3.5) we have

    α1=m+1+m2+6m+12.
    α2=m+1m2+6m+12.

    So we have,

    specA=(0m+1+m2+6m+12m+1m2+6m+12.m111) (3.6)

    Using Eq (3.6) we have

    specA=(0m+1+m2+6m+12m+1m2+6m+12.m111).

    Since MI(Splm(G))=MI(G)A then by Proposition 1.1, we have

    MI(Splm(G))=Maxni=1|(specA))ηi|
    MI(Splm(G))=Maxni=1|ηi|[m+1+m2+6m+12]
    MI(Splm(G))=MI(G)(m+1+m2+6m+12).

    In the following corollaries, we obtain the minimum degree spectral radii of splitting graphs of Cn, Kn, Cn,n and crown graph.

    Corollary 7. If n3 and G is a Cn graph, where Cn is cycle graph on n vertices, then

    MI(Splm(Cn))=4(m+1+m2+6m+12).

    Proof. If G is a cycle graph Cn(n3), then MI(Cn)=4. Since cycle graph is 2-regular graph so using Theorem 6 we get the required result.

    Corollary 8. If G is a Kn graph, where Kn is complete graph on n vertices, then

    MI(Splm(Kn))=(n1)2(m+1+m2+6m+12).

    Proof. If G is a complete graph on n vertices, then MI(Kn)=(n1)2. Since complete graph is n1-regular graph so using Theorem 6 we get the required result.

    Corollary 9. If G is a complete bipartite graph Kn,n, then

    MI(Splm(Kn,n))=(n)2(m+1+m2+6m+12).

    Proof. If G is a complete bipartite graph Kn,n, then MI(Kn,n)=(n)2. Using Theorem 6 we get the required result.

    Corollary 10. If G is a crown graph on 2n vertices, then

    MI(Splm(G))=(n1)2(m+1+m2+6m+12).

    Proof. If G is a crown graph on 2n vertices, then MI(G)=(n1)2. Using Theorem 6 we get the required result.

    In this part we relate maximum degree spectral radius and minimum degree spectral radius of generalized Shadow graph with original graph G. Here again we emphasis that G is any regular graph.

    Theorem 11. Let G be any n-regular graph and M(Shm(G)) is themaximum degree spectral radius of m-shadow graph G, then

    M(Shm(G))=M(G)((m)2).

    Proof. Maximum degree matrix is given by

    M(G)={max(di,dj),ifviandvjareadjacent,0,elsewhere.}

    Then M(Shm(G)) can be written in block matrix form as follows:

    M(Shm(G))={(m)Miandj.}
    =M{miandj.}

    Let

    A={miandj.}m

    Now we compute the eigenvalues of A Since matrix A is of rank one, so A has one non-zero eigenvalue, say α1=(m)2. So we have,

    specA=(0(m)2m11) (4.1)

    Using Eq (4.1) we have

    specA=(0(m)2m11).

    Since M(Shm(G))=M(G)A then by Proposition 1.1, we have

    M(Shm(G))=Maxni=1|(specA))ηi|
    M(Shm(G))=Maxni=1|ηi|[(m)2]
    M(Shm(G))=M(G)((m)2).

    In the following corollaries, we obtain the maximum degree spectral radii of shadow graphs of Cn, Kn, Cn,n and crown graph.

    Corollary 12. If n3 and G is a Cn graph, where Cn is cycle graph on n vertices, then

    M(Shm(Cn))=4((m)2).

    Proof. If G is a cycle graph Cn(n3), then M(Cn)=4. Since cycle graph is 2-regular graph so using Theorem 11 we get the required result.

    Corollary 13. If G is a Kn graph, where Kn is complete graph on n vertices, then

    M(Shm(Kn))=(n1)2((m)2).

    Proof. If G is a complete graph on n vertices, then M(Kn)=(n1)2. Since complete graph is n1-regular graph so using Theorem 11, we get the required result.

    Corollary 14. If G is a complete bipartite graph Kn,n, then

    M(Shm(Kn,n))=(n)2((m)2).

    Proof. If G is a complete bipartite graph Kn,n, then M(Kn,n)=(n)2. Using Theorem 11, we get the required result.

    Corollary 15. If G is a crown graph on 2n vertices, then

    M(Shm(G))=(n1)2((m)2).

    Proof. If G is a crown graph on 2n vertices, then M(G)=(n1)2. Using Theorem 11, we get the required result.

    Theorem 16. Let G be any n-regular graph and MI(Shm(G)) is theminimum degree spectral radius of m-shadow graph G, then

    MI(Shm(G))=(m)2MI(G).

    Proof. Minimum degree matrix is given by

    MI={min(di,dj),ifviandvjareadjacent,0,elsewhere.}

    Then MI(Shm(G)) can be written in block matrix form as follows:

    MI(Shm(G))={(m)MIiandj.}
    =MI{miandj.}

    Let

    A={miandj.}m

    Now we compute the eigenvalues of A Since matrix A is of rank one, so A has one non-zero eigenvalue, say α1=(m)2. So we have,

    specA=(0(m)2m11) (4.2)

    Using Eq (4.2) we have

    specA=(0(m)2m11).

    Since MI(Shm(G))=MI(G)A then by Proposition 1.1, we have

    MI(Shm(G))=Maxni=1|(specA))ηi|
    MI(Shm(G))=Maxni=1|ηi|[(m)2]
    MI(Shm(G))=MI(G)((m)2).

    In the following corollaries, we obtain the minimum degree spectral radii of shadow graphs of Cn, Kn, Cn,n and crown graph.

    Corollary 17. If n3 and G is a Cn graph, where Cn is cycle graph on n vertices, then

    MI(Shm(Cn))=4((m)2).

    Proof. If G is a cycle graph Cn(n3), then MI(Cn)=4. Since cycle graph is 2-regular graph so using Theorem 16 we get the required result.

    Corollary 18. If G is a Kn graph, where Kn is complete graph on n vertices, then

    MI(Shm(Kn))=(n1)2((m)2).

    Proof. If G is a complete graph on n vertices, then MI(Kn)=(n1)2. Since complete graph is n1-regular graph so using Theorem 16 we get the required result.

    Corollary 19. If G is a complete bipartite graph Kn,n, then

    MI(Shm(Kn,n))=(n)2((m)2).

    Proof. If G is a complete bipartite graph Kn,n, then MI(Kn,n)=(n)2. Using Theorem 16, we get the required result.

    Corollary 20. If G is a crown graph on 2n vertices, then

    MI(Shm(G))=(n1)2((m)2).

    Proof. If G is a crown graph on 2n vertices, then MI(G)=(n1)2. Using Theorem 16 we get the required result.

    The spectral radius of graph has vast range of applications in computer related areas. It also connects graph theory and chemistry. In this article we have related the spectral radii of the generalized shadow and splitting graph of any regular graph with spectral radius of the given graph. In particular we have proved that the Spectral Radius of the new graph is a multiple of spectral radius of the given regular graph.

    The authors thank the National Key Research and Development Program under Grant 2018YFB0904205, Science and Technology Bureau of ChengDu (2020-YF09-00005-SN), Sichuan Science and Technology program (2021YFH0107) and Erasmus + SHYFTE Project (598649-EPP-1-2018-1-FR-EPPKA2-CBHE-JP).

    The authors declare that there is no conflict of interest.



    [1] I. Gutman, The energy of a graph, Ber. Math. Statist. Sekt. Forschungsz. Graz, 103 (1978), 1-22. https://doi.org/10.1002/9783527627981
    [2] D. Cvetkovic, M. Doob, H. Sachs, Spectra of Graphs Theory and Applications, Academic Press, 1982.
    [3] D. Cvetkovic, P. Rowlinson, The largest eigenvalue of a graph, Linear Multilinear Algebra, 28 (1990), 3-33. https://doi.org/10.1080/03081089008818026 doi: 10.1080/03081089008818026
    [4] Y. Hong, Upper bounds of the spectral radius of graphs in terms of genus, J. Combin. Theory Ser., 74 (1998), 153-159. https://doi.org/10.1006/jctb.1998.1837 doi: 10.1006/jctb.1998.1837
    [5] R. A. Brualdi, E. S. Solheid, On the spectral radius of complementary acyclic matrices of zeros and ones, SIAM J. Algebraic Discrete Methods, 7 (1986), 265-272. https://doi.org/10.1137/0607030 doi: 10.1137/0607030
    [6] M. Fiedler, V. Nikiforov, Spectral radius and Hamiltonicity of graphs, Linear Algebra Appl., 432 (2010), 2170-2173. https://doi.org/10.1016/j.laa.2009.01.005 doi: 10.1016/j.laa.2009.01.005
    [7] M. Lu, H. Liu, F. Tian, Spectral radius and Hamiltonian graphs, Linear Algebra Appl., 437 (2012), 1670-1674. https://doi.org/10.1016/j.laa.2012.05.021 doi: 10.1016/j.laa.2012.05.021
    [8] V. Nikiforov, More spectral bounds on the clique and independence numbers, J. Combin. Theory Ser., 99 (2009), 819-826. https://doi.org/10.1016/j.jctb.2009.01.003 doi: 10.1016/j.jctb.2009.01.003
    [9] V. Nikiforov, The spectral radius of graphs without paths and cycles of specified length, Linear Algebra Appl., 432 (2010), 2243-2256. https://doi.org/10.1016/j.laa.2009.05.023 doi: 10.1016/j.laa.2009.05.023
    [10] W. Yuan, B. Wang, M. Zhai, On the spectral radii of graphs without given cycles, Electron. J. Linear Algebra, 23 (2012), 599-606. https://doi.org/10.13001/1081-3810.1544 doi: 10.13001/1081-3810.1544
    [11] M. Zhai, B. Wang, Proof of a conjecture on the spectral radius of C4-free graphs, Linear Algebra Appl., 437 (2012), 1641-1647. https://doi.org/10.1016/j.laa.2012.05.006 doi: 10.1016/j.laa.2012.05.006
    [12] S. Butler, F. Chung, Small spectral gap in the combinatorial Laplacian implies Hamiltonian, Ann. Comb., 13 (2010), 403-412. https://doi.org/10.1007/s00026-009-0039-4 doi: 10.1007/s00026-009-0039-4
    [13] J. V. Heuvel, Hamilton cycles and eigenvalues of graphs, Linear Algebra Appl., 228 (1995), 723-730. https://doi.org/10.1016/0024-3795(95)00254-O doi: 10.1016/0024-3795(95)00254-O
    [14] M. Krivelevich, B. Sudakov, Sparse pseudo-random graphs are Hamiltonian, J. Graph Theory, 42 (2003), 17-33. https://doi.org/10.1002/jgt.10065 doi: 10.1002/jgt.10065
    [15] B. Mohar, A domain monotonicity theorem for graphs and hamiltonicity, Discrete Appl. Math., 36 (1992), 169-177. https://doi.org/10.1016/0166-218X(92)90230-8 doi: 10.1016/0166-218X(92)90230-8
    [16] B. Zhou, Signless Laplacian spectral radius and Hamiltonicity, Linear Algebra Appl., 432 (2010), 566-570. https://doi.org/10.1016/j.laa.2009.09.004 doi: 10.1016/j.laa.2009.09.004
    [17] R. A. Horn, C. R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, 1991. https://doi.org/10.1017/CBO9780511840371
    [18] F. R. Gantmacher, The Theory of Matrices, Chelsea, 1959.
    [19] R. Balakrishnan, The energy of a graph, Lin. Algebra Appl., 387 (2004), 287-295. https://doi.org/10.1016/j.laa.2004.02.038
    [20] R. B. Bapat, S. Pati, Energy of a graph is never an odd integer, Bull. Kerala Math. Assoc., 1 (2004), 129-132. https://doi.org/10.2298/AADM0801118P doi: 10.2298/AADM0801118P
    [21] S. Pirzada, S. Gutman, Energy of a graph is never the square root of an odd integer, Appl. Anal. Discr. Math., 2 (2008), 118-121. https://doi.org/10.2298/AADM0801118P doi: 10.2298/AADM0801118P
    [22] O. Jones, Spectra of simple graphs, Whitman College, 13 (2013), 1-20.
    [23] S. Meenaksh, S. Lavanya, A Survey on Energy of Graphs, Ann. Pure Appl. Math., 8 (2014), 183-191.
    [24] V. Nikiforov, The energy of graphs and matrices, J. Math. Anal. Appl., 326 (2007), 1472-1475. https://doi.org/10.1016/j.jmaa.2006.03.072 doi: 10.1016/j.jmaa.2006.03.072
    [25] K. Samir, Vaidya, M. P. Kalpesh, Some new results on energy of graphs, MATCH Common. Math Comput. Chem, 77 (2017), 589-594.
    [26] K. Samir, Vaidya, M. P. Kalpesh, On energy m-splitting and m-shadow graphs, Far East J. Of Math. Sci., 102 (2017), 1571-1578. https://doi.org/10.17654/MS102081571 doi: 10.17654/MS102081571
    [27] J. B. Liu, M. Munir, A. Yousaf, A. Naseem, K. Ayub, Distance and adjacency energies of multi-level wheel networks, Mathematics, 7 (2019), 43. https://doi.org/10.3390/math7010043 doi: 10.3390/math7010043
    [28] Z. Q. Chu, M. Munir, A. Yousaf, M. I. Qureshi, J, B. Liu, Laplacian and signless laplacian spectra and energies of multi-step wheels, Math. Biosci. Eng., 17 (2020), 3649-3659. https://doi.org/10.3934/mbe.2020206 doi: 10.3934/mbe.2020206
    [29] J. B. Liu, X. F. Pan, F. T. Hu, F. HuF, On Asymptotic Laplacian-energy-like invariant of lattices, Appl. Math. Comput., 253 (2015), 205-214. https://doi.org/10.1016/j.amc.2014.12.035 doi: 10.1016/j.amc.2014.12.035
    [30] S. M. Hosamani, H. S. Ramane, On degree sum energy of a graph, Eur. J. Pure Appl. Math., 9 (2016), 340-345.
    [31] B. Basavanagoud, E. Chitra, On degree square sum energy of graphs, Int. J. Math. Its Appl., 6 (2018), 193-205.
    [32] N. J. Rad, A. Jahanbani, I. Gutman, Zagreb energy and Zagreb Estrada index of graphs, MATCH Commun. Math. Comput. Chem., 79 (2018), 371-386.
    [33] I. Gutman, B. Furtula, On graph energies and their application, Bulletin (Acadmie Serbe Des Sciences Et Des Arts. Classe Des Sciences Mathmatiques Et Naturelles. SciencesMath matiques), 44 (2019), 29-45.
    [34] X. Li, Y. Shi, I. Gutman, Graph Energy, Springer, New York, 2012. https://doi.org/10.1007/978-1-4614-4220-2
    [35] R. Balakrishnan, K. Ranganathan, A Textbook of Graph Theory, Springer, 2000. https://doi.org/10.1007/978-1-4419-8505-7
    [36] R. J. Wilson, Introduction to Graph Theory, Oliver and Boyd, 1972.
    [37] J. J. Sylvester, Chemistry and algebra, Nature, 17 (1877), 284. https://doi.org/10.1038/017284a0
    [38] A. T. Balaban, Applications of graph theory in chemistry, J. Chem. Inf. Comput. Sci., 25 (1985), 334-343. https://doi.org/10.1021/ci00047a033 doi: 10.1021/ci00047a033
    [39] M. RandiC, On characterization of molecular branching, J. Amer. Chem. Soc., 97 (1975), 6609-6615. https://doi.org/10.1021/ja00856a001 doi: 10.1021/ja00856a001
    [40] K. Yuge, Extended configurational polyhedra based on graph representation for crystalline solids, Trans. Material Res. Soc., 43 (2018), 233-236. https://doi.org/10.14723/tmrsj.43.233 doi: 10.14723/tmrsj.43.233
    [41] A. Dhanalakshmi, K. S. Rao, K. Sivakumar, Characterization of a-cyclodextrin using adjacency and distance matrix, Indian J. Sci., 12 (2015), 78-83.
    [42] J. Praznikar, M. Tomic, D. Turk, Validation and quality assessment of macromolecular structures using complex network analysis, Sci. Rep., 9 (2019), 1678. https://doi.org/10.1038/s41598-019-38658-9 doi: 10.1038/s41598-019-38658-9
    [43] H. Wu, Y. Zhang, W. Chen, Z. Mu, Comparative analysis of protein primary sequences with graph energy, Phys. A, 437 (2015), 249-262. https://doi.org/10.1016/j.physa.2015.04.017 doi: 10.1016/j.physa.2015.04.017
    [44] L. D. Paola, G. Mei, A. D. Venere, A. Giuliani, Exploring the stability of dimers through protein structure topology, Curr. Protein Peptide Sci., 17 (2016), 30-36. https://doi.org/10.2174/1389203716666150923104054 doi: 10.2174/1389203716666150923104054
    [45] D. Sun, C. Xu, Y. Zhang, A novel method of 2D graphical representation for proteins and its application, MATCH Commun. Math. Comput. Chem., 75 (2016), 431-446.
    [46] L. Yu, Y. Z. Hang, I. Gutman, Y. Shi, M. Dehmer, Protein sequence comparison based on physicochemical properties and positionfeature energy matrix, Sci. Rep., 7 (2017), 46237. https://doi.org/10.1038/srep46237 doi: 10.1038/srep46237
    [47] A. Giuliani, S. Filippi, M. Bertolaso, Why network approach can promote a new way of thinking in biology, Front. Genet., 5 (2014), 83. https://doi.org/10.3389/fgene.2014.00083 doi: 10.3389/fgene.2014.00083
    [48] J. Jiang, R. Zhang, L. Guo, W. Li, X. Cai, Network aggregation process in multilayer air transportation networks, Chin. Phys. Lett., 33 (2016), 108901. https://doi.org/10.1088/0256-307X/33/10/108901 doi: 10.1088/0256-307X/33/10/108901
    [49] T. A. Shatto, E. K. C. Etinkaya, Variations in graph energy: A measure for network resilience, in 2017 9th International Workshop on Resilient Networks Design and Modeling (RNDM), (2017), 1-7. https://doi.org/10.1109/RNDM.2017.8093019
    [50] M. Akram, S. Naz, Energy of Pythagorean fuzzy graphs with applications, Mathematics, 6 (2018), 136. https://doi.org/10.3390/math6080136 doi: 10.3390/math6080136
    [51] M. Gao, E. P. Lim, D. Lo, Network Data Mining and Analysis, World Scientific, 4 (2018), 1-4. https://doi.org/10.1142/9789813274969-0001
    [52] A. Pugliese, R. Nilchiani, Complexity analysis of fractionated spacecraft architectures, in AIAA SPACE and Astronautics Forum and Exposition, (2017), 5118-5126. https://doi.org/10.2514/6.2017-5118
    [53] C. Adiga, M. Smitha, On maximum degree energy of a graph, Int. J. Contemp. Math. Sci., 4 (2009), 385-396.
    [54] B. Basavanagoud, P. Jakkannavar, Minimum degree energy pf graphs, Electron. J. Math. Anal. Appl., 7 (2019), 230-243.
    [55] S. Alikhani, N. Ghanbari, Randic energy of specific graphs, Appl. Math and Comp, 269 (2015), 722-730. https://doi.org/10.1016/j.amc.2015.07.112 doi: 10.1016/j.amc.2015.07.112
    [56] M. R. Oboudi, Energy and seidel energy of graphs, MATCH Commun. Math. Comput. Chem., 75 (2016), 291-303.
  • This article has been cited by:

    1. Ahmad Bilal, Mobeen Munir, Randic and reciprocal randic spectral radii and energies of some graph operations, 2022, 10641246, 1, 10.3233/JIFS-221938
    2. Ahmad Bilal, Muhammad Mobeen Munir, ABC energies and spectral radii of some graph operations, 2022, 10, 2296-424X, 10.3389/fphy.2022.1053038
    3. Ahmad Bilal, Muhammad Mobeen Munir, Muhammad Imran Qureshi, Muhammad Athar, ISI spectral radii and ISI energies of graph operations, 2023, 11, 2296-424X, 10.3389/fphy.2023.1149006
    4. Linlin Wang, Sujuan Liu, Han Jiang, Rainbow connections of bioriented graphs, 2024, 10, 24058440, e31426, 10.1016/j.heliyon.2024.e31426
    5. A. R. Nagalakshmi, A. S. Shrikanth, G. K. Kalavathi, K. S. Sreekeshava, The Degree Energy of a Graph, 2024, 12, 2227-7390, 2699, 10.3390/math12172699
    6. Muhammad Mobeen Munir, Urwah Tul Wusqa, Albertson (Alb) spectral radii and Albertson (Alb) energies of graph operation, 2023, 11, 2296-2646, 10.3389/fchem.2023.1267291
    7. Caicai Feng, Muhammad Farhan Hanif, Muhammad Kamran Siddiqui, Mazhar Hussain, Nazir Hussain, On analysis of entropy measure via logarithmic regression model for 2D-honeycomb networks, 2023, 138, 2190-5444, 10.1140/epjp/s13360-023-04547-4
    8. Ahmad Bilal, Muhammad Mobeen Munir, SDD Spectral Radii and SDD Energies of Graph Operations, 2024, 1007, 03043975, 114651, 10.1016/j.tcs.2024.114651
    9. Xueliang Li, Ruiling Zheng, Extremal results on the spectral radius of function-weighted adjacency matrices, 2025, 373, 0166218X, 204, 10.1016/j.dam.2025.04.060
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2888) PDF downloads(138) Cited by(9)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog