The hyper edge-Wiener index of corona product of graphs

Document Type: Research Paper

Authors

1 Tarbiat Modares University

2 Department of Mathematics, Tarbiat Modares University, P. O. Box 14115-137, Tehran

Abstract

‎Let $G$ be a simple connected graph‎. ‎The edge-Wiener index $W_e(G)$‎ ‎is the sum of all distances between edges in $G$‎, ‎whereas the hyper‎ ‎edge-Wiener index $WW_e(G)$ is defined as ‎ ‎$W{W_e}(G) = {\frac{1}{2}}{W_e}(G)‎ + ‎{\frac{1}{2}} {W_e^{2}}(G)$‎, ‎where $ {W_e^{2}}(G)= \sum\limits_{\left\{ {f,g} \right\}‎ ‎\subseteq E(G)} {d_e^2(f,g)}$‎. ‎In this paper‎, ‎we present explicit formula for the hyper edge-Wiener‎ ‎index of corona product of two graphs‎. ‎Also‎, ‎we use it to‎ ‎determine the hyper edge-Wiener index of some chemical graphs‎.

Keywords

Main Subjects


Y‎. ‎Alizadeh‎, ‎A‎. ‎Iranmanesh‎, ‎T‎. ‎Doslic and M‎. ‎Azari (2014). ‎The edge Wiener index of suspensions‎, ‎bottlenecks‎, ‎and thorny graphs. Glas‎. ‎Mat‎. ‎Ser‎. ‎III. 49 (69), 1-12
P‎. ‎Dankelmann‎, ‎I‎. ‎Gutman‎, ‎S‎. ‎Mukwembi and H‎. ‎C‎. ‎Swart (2009). ‎The edge Wiener index of a graph. Discrete Math.. 309 (10), 3452-3457
J‎. ‎Devillers and A‎. ‎Balaban (1999). Topological Indices and Related Descriptions in QSAR and QSPR. ‎Gordon and Breech‎, ‎Amsterdam.
A‎. ‎A‎. ‎Dobrynin‎, ‎R‎. ‎Entringer and I‎. ‎Gutman (2001). ‎Wiener index of trees‎: ‎theory and applications. Acta Appl‎. ‎Math.. 66, 211-249
A‎. ‎A‎. ‎Dobrynin‎, ‎I‎. ‎Gutman‎, ‎S‎. ‎Klavzar and P‎. ‎Zigert (2002). ‎Wiener index of hexagonal systems. Acta Appl‎. ‎Math.. 72, 247-294
I‎. ‎Gutman and N‎. ‎Trinajstic (1972). ‎Graph theory and molecular orbitals‎. ‎Total $pi$-electron energy of alternant hydrocarbons. Chem‎. ‎Phys‎. ‎Lett.. 17, 535-538
I‎. ‎Gutman (1997). ‎A property of the Wiener number and its modifications. Indian J‎. ‎Chem.. 36A, 128-132
W‎. ‎Imrich and S‎. ‎Klavzar (2000). Product graphs‎: ‎structure and recognition. ‎John Wiley & Sons‎, ‎New York‎, ‎USA.
A‎. ‎Iranmanesh‎, ‎I‎. ‎Gutman‎, ‎O‎. ‎Khormali and A‎. ‎Mahmiani (2009). ‎The edge versions of Wiener index. MATCH Commun‎. ‎Math‎. ‎Computt‎. ‎Chem.. 61 (3), 663-672
A‎. ‎Iranmanesh‎, ‎A‎. ‎S‎. ‎Kafrani and O‎. ‎Khormali (2011). ‎A new version of hyper-Wiener index. MATCH Commun‎. ‎Math‎. ‎Computt‎. ‎Chem.. 65 (1), 113-122
M‎. ‎H‎. ‎Khalifeh‎, ‎H‎. ‎Yousefi-Azari‎, ‎A‎. ‎R‎. ‎Ashrafi and S‎. ‎G‎. ‎Wagner (2009). ‎Some new results on distance-based graph invariants. European J‎. ‎Combin.. 30, 1149-1163
D‎. ‎J‎. ‎Klein‎, ‎I‎. ‎Lukovits and I‎. ‎Gutman (1995). ‎On the definition of Hyper-Wiener index for cycle-containing structures. J‎. ‎Chem‎. ‎Inf‎. ‎Comput‎. ‎Phys‎. ‎Chem‎. ‎Sci.. 35, 50-52
D‎. ‎J‎. ‎Klein‎, ‎T‎. ‎Doslic and D‎. ‎Bonchev (2007). ‎Vertex-weightings for distance moments and thorny graphs. Discrete Appl‎. ‎Math.. 155, 2294-2302
A‎. ‎Milicevic and N‎. ‎Trinajstic (2006). Combinatorial enumeration in chemistry‎, ‎chemical modelling‎: ‎Applications ‎ ‎‎nd theory. ‎(A‎. ‎Hincliffe‎, ‎Ed.)‎, ‎RSC Publishing‎, ‎Cambridge. , 405-469
S‎. ‎Nikolic‎, ‎N‎. ‎Trinajstic and Z‎. ‎Mihalic (1995). ‎The‎ ‎Wiener index‎: ‎Development and applications. Croat‎. ‎Chem‎. ‎Acta.. 68, 105-129
M‎. ‎Randic (1993). ‎Novel molecular descriptor for structure property studies. Chem‎. ‎Phys‎. ‎Lett.. 211, 478-483
A‎. ‎Soltani‎, ‎A‎. ‎Iranmanesh and Z‎. ‎A‎. ‎Majid (2013). ‎The edge Wiener type topological indices. Util‎. ‎Math.. 91, 87-98
R‎. ‎Todeschini and V‎. ‎Consonni (2000). Handbook of molecular descriptors. ‎Wiley‎, ‎Weinheim.
H‎. ‎Wiener (1947). ‎Structural determination of paraffin boiling points. J‎. ‎Am‎. ‎Chem‎. ‎Soc.. 69, 17-20