Further results on resistance distance and Kirchhoff index in electric networks
Summary: 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. Let \(Q(G)\) be the graph obtained from \(G\) by inserting a new vertex into every edge of \(G\) and by joining by edges those pairs of these new vertices which lie on adjacent edges of \(G\). The set of such new vertices is denoted by \(I(G)\). The \(Q\)-vertex corona of \(G_1\) and \(G_2\), denoted by \(G_1 \odot_Q G_2\), is the graph obtained from vertex disjoint \(Q(G_1)\) and \(\left|V \left(G_1\right)\right|\) copies of \(G_2\) by joining the \(i\)th vertex of \(V(G_1)\) to every vertex in the \(i\)th copy of \(G_2\). The \(Q\)-edge corona of \(G_1\) and \(G_2\), denoted by \(G_1 \ominus_Q G_2\), is the graph obtained from vertex disjoint \(Q(G_1)\) and \(\left|I \left(G_1\right)\right|\) copies of \(G_2\) by joining the \(i\)th vertex of \(I(G_1)\) to every vertex in the \(i\)th copy of \(G_2\). The objective of the present work is to obtain the resistance distance and Kirchhoff index for composite networks such as \(Q\)-vertex corona and \(Q\)-edge corona networks.
scientific article; zbMATH DE number 7025596
scientific article