Abstract
In electric circuit theory, it is of great interest to compute the effective resistance between any pairs of vertices of a network, as well as the Kirchhoff index. LetQ(G)be the graph obtained fromGby inserting a new vertex into every edge ofGand by joining by edges those pairs of these new vertices which lie on adjacent edges ofG. The set of such new vertices is denoted byI(G). TheQ-vertex corona ofG1andG2, denoted byG1⊙QG2, is the graph obtained from vertex disjointQ(G1)andVG1copies ofG2by joining theith vertex ofV(G1)to every vertex in theith copy ofG2. TheQ-edge corona ofG1andG2, denoted byG1⊖QG2, is the graph obtained from vertex disjointQ(G1)andIG1copies ofG2by joining theith vertex ofI(G1)to every vertex in theith copy ofG2. The objective of the present work is to obtain the resistance distance and Kirchhoff index for composite networks such asQ-vertex corona andQ-edge corona networks.
Highlights
For decades we want to know what a graph looks like
It is of great interest to compute the effective resistance between any pairs of vertices of a network, as well as the Kirchhoff index
Klein and Randic [1] introduced a new distance function named resistance distance based on electric network theory; the resistance distance between vertices Vi and Vj, denoted by rG(Vi, Vj), is defined to be the effective electrical resistance between them if each edge of G is replaced by a unit resistor
Summary
For decades we want to know what a graph looks like. We want to reveal the principles of the networks behaviour covered by their complex topology and dynamics. It is known that resistance distances in a connected graph G can be obtained from any {1}-inverse of G [7, 8]. Liu et al [5] gave the resistance distance and Kirchhoff index of Rvertex join and R-edge join of two graphs. Motivated by these works, in this paper we will work on two different composite networks: Q-vertex corona and Q-edge corona networks. We will follow the techniques used in [4] but in a slightly different method in order to obtain the effective resistances and the Kirchhoff indexes of Q-vertex corona and Q-edge corona networks in terms of the same parameters on the factors
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