On Vertex-Edge and Edge-Vertex Connectivity Indices of Graphs

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DOI: 10.46793/KgJMat2402.225P


Let G be a graph with vertex set V (G) and edge set E(G). The vertex-edge degree of the vertex v, dGe(v), equals to the number of different edges that are incident to any vertex from the open neighborhood of v. Also, the edge-vertex degree of the edge e = uv, dGv(e), equals to the number of vertices of the union of the open neighborhood of u and v. In this paper, the vertex-edge connectivity index, ϕv, and the edge-vertex connectivity index, ϕe, of a graph G were introduced. These are defined as ϕv(G) = vV (G)dGe(v)dG(v) and ϕe(G) = e=uvE(G)dG(e)dGv(e), where dG(v) is the degree of a vertex v V (G) and dG(e) is the number of edges in E(G) that are adjacent to e. In this paper, we will study the main properties of ϕv(G), ϕe(G) and establish some upper and lower bounds for them. The numbers ϕv and ϕe for titania nanotubes are also computed.


Vertex-edge degree, edge-vertex degree, vertex-edge connectivity index, edge-vertex connectivity index.


[1]   A. R. Ashrafi, T. Došlić and A. Hamzeh, The Zagreb coindices of graph operations, Discrete Appl. Math. 158 (2010), 1571–1578. https://doi.org/10.1016/j.dam.2010.05.017

[2]   B. Basavanagoud and E. Chitra, On the leap Zagreb indices of generalized xyz-point-line transformation graphs Txyz(G) when z = 1, Int. J. Math. Combin. 2 (2018), 44–66.

[3]   B. Borovićanin, K. C. Das, B. Furtula and I. Gutman, Bounds for Zagreb indices, MATCH Commun. Math. Comput. Chem. 78(1) (2017), 17–100.

[4]   R. Boutrig, M. Chellali, T. W. Haynes and S. T. Hedetniemi, Vertex-edge domination in graphs, Aequationes Math. 90(2) (2016), 355–366. https://doi.org/10.1007/s00010-015-0354-2

[5]   M. Cancan and M. S. Aldemir, On ve-degree and ev-degree Zagreb index of titania nanotubes, American Journal of Chemical Engineering 5(6) (2017), 163–168. https://doi.org/10.11648/j.ajche.20170506.18

[6]   M. Chellali, T. W. Haynes, S. T. Hedetniemi and T. M. Lewis, On ve-degrees and ev-degrees in graphs, Discrete Math. 340 (2017), 31–38. https://doi.org/10.1016/j.disc.2016.07.008

[7]   K. C. Das and I. Gutman, Some properties of the second Zagreb index, MATCH Commun. Math. Comput. Chem. 52 (2004), 103–112.

[8]   S. Ediz, On ve-degree molecular topological properties of silicate and oxygen networks, Int. J. Comput. Sci. Math. 9(1) (2018), 1–12. https://doi.org/10.1504/IJCSM.2018.090730

[9]   S. Ediz, Predicting some physicochemical properties of octane isomers: A topological approach using evdegree and ve-degree Zagreb indices, International Journal of Systems Science and Applied Mathematics. 2(5) (2017), 87–92. https://doi.org/10.11648/j.ijssam.20170205.12

[10]   M. Eliasi, A. Iranmanesh and I. Gutman, Multiplicative versions of first Zagreb index, MATCH Commun. Math. Comput. Chem. 68 (2012), 217–230.

[11]    M. R. Farahani, M. K. Jamil and M. Imran, Vertex PIv topological index of Titania nanotubes, Appl. Math. Nonlinear Sci. 1 (2016), 170–175. https://doi.org/10.21042/AMNS.2016.1.00013

[12]   B. Furtula and I. Gutman, A forgotten topological index, J. Math. Chem. 53 (2015), 1184–1190. https://doi.org/10.1007/s10910-015-0480-z

[13]   I. Gutman, Multiplicative Zagreb indices of trees, Bull. Int. Math. Virtual Inst. 18 (2011), 17–23.

[14]   I. Gutman, B. Ruščic, N. Trinajstić and C. F. Wilcox, Graph theory and molecular orbitals, XII. Acyclic polyenes, J. Chem. Phys. 62 (1975), 3399–3405. https://doi.org/10.1063/1.430994

[15]   I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), 535–538. https://doi.org/10.1016/0009-2614(72)85099-1

[16]   I. Gutman, B. Furtula, Z. K. Vukićević and G. Popivoda, On Zagreb indices and coindices, MATCH Commun. Math. Comput. Chem. 74 (2015), 5–16.

[17]   F. Harary, Graph Theory, Addison-Wesley Publishing Co., Reading, Mass. Menlo Park, London, 1969.

[18]   J. Lewis, S. T. Hedetniemi, T. W. Haynes and G. H. Fricke, Vertex-edge domination, Util. Math. 81 (2010), 193–213.

[19]   J. Lewis, Vertex-edge and edge-vertex parameters in graphs, Ph.D. Thesis, Clemson University, 2007.

[20]   Y. Z. Li, N. H. Lee, E. G. Lee, J. S. Song and S. J. Kim, The characterization and photocatalytic properties of mesoporous rutile TiO2 powder synthesized through cell assembly of nanocrystals, Chem. Phys. Lett. 389 (2004), 124–128. https://doi.org/10.1016/j.cplett.2004.03.081

[21]   K. W. Peters, Theoretical and algorithmic results on domination and connectivity (Nordhaus-Gaddum, Gallai type results, max-min relationships, linear time, series-parallel), Ph.D. Thesis, Clemson University, 1986.

[22]   A. M. Naji, N. D. Soner and I. Gutman, On leap Zagreb indices of graphs, Commun. Comb. Optim. 2(2) (2017), 99–117. https://doi.org/10.22049/CCO.2017.25949.1059

[23]   A. M. Naji and N. D. Soner, The first leap Zagreb index of some graph opertations, International Journal of Applied Graph Theory 2(1) (2018), 7–18.

[24]   S. Nikolić, G. Kovačević, A. Milićević and N. Trinajstić, The Zagreb indices 30 years after, Croatica Chemica Acta 76 (2003), 113–124.

[25]   P. Shiladhar, A. M. Naji and N. D. Soner, Leap Zagreb indices of some wheel related graphs, J. Comput. Math. Sci. 9(3) (2018), 221–231.

[26]   P. Shiladhar, A. M. Naji and N. D. Soner, Computation of leap Zagreb indices of some windmill graphs, International Journal of Mathematics and its Applications 6(2-B) (2018), 183–191.

[27]   B. Sahin and S. Ediz, On ev-degree and ve-degree topological indices, Iranian Journal of Mathematical Chemistry 9(4) (2018), 263–277. https://doi.org/10.22052/IJMC.2017.72666.1265

[28]   N. D. Soner and A. M. Naji, The k-distance neighborhood polynomial of a graph, Int. J. Math. Comput. Sci. 3(9) (2016), 2359–2364.

[29]   R. Todeschini and V. Consonni, New local vertex invariants and molecular descriptors based on functions of the vertex degrees, MATCH Commun. Math. Comput. Chem. 64 (2010), 359–372.

[30]   K. Xu and H. Hua, A unified approach to extremal multiplicative Zagreb indices for trees, unicyclic and bicyclic graphs, MATCH Commun. Math. Comput. Chem. 68 (2012), 241–256.

[31]   K. Xu, K. C. Das and K. Tang, On the multiplicative Zagreb coindex of graphs, Opuscula Math. 33(1) (2013), 197–210. http://dx.doi.org/10.7494/OpMath.2013.33.1.191

[32]   S. Yamaguchi, Estimating the Zagreb indices and the spectral radius of triangle and quadrangle-free connected graphs, Chemical Physics Letters 458(4) (2008), 396–398. https://doi.org/10.1016/j.cplett.2008.05.009