Resistance distance in kk-coalescence of certain graphs

Any graph can be considered as a network of resistors, each of which has a resistance of $1 \Omega.$ The resistance distance $r_{ij}$ between a pair of vertices $i$ and $j$ in a graph is defined as the effective resistance between $i$ and $j$. This article deals with the resistance distance in the $k$-coalescence of complete graphs. We also present its results in connection with the Kemeny's constant, Kirchhoff index, additive degree-Kirchhoff index, multiplicative degree-Kirchhoff index and mixed degree-Kirchhoff index. Moreover, we obtain the resistance distance in the $k$-coalescence of a complete graph with particular graphs. As an application, we provide the resistance distance of certain graphs such as the vertex coalescence of a complete bipartite graph with a complete graph, a complete bipartite graph with a star graph, the windmill graph, pineapple graph, etc

Extremal polygonal chains with respect to the Kirchhoff index

The Kirchhoff index is defined as the sum of resistance distances between all pairs of vertices in a graph. This index is a critical parameter for measuring graph structures. In this paper, we characterize polygonal chains with the minimum Kirchhoff index, and characterize even (odd) polygonal chains with the maximum Kirchhoff index, which extends the results of \cite{45}, \cite{14} and \cite{2,13,44} to a more general case.Comment: 13 pages. arXiv admin note: substantial text overlap with arXiv:2209.1026

On the zeta Kirchhoff index of several graph transformations

In this paper, we first derived the Ihara zeta function, complexity and zeta Kirchhoff index of the k-th semitotal point graph (of regular graphs), a construction by Cui and Hou [5], where we create triangles for every edge in the original graph. Then, we extend the construction to the creation of equilaterals and polygons. We also derived the zeta Kirchhoff indices for numerous graph transformations on regular graphs, and some selected families of graphs. At the end, a data summary is provided for enumeration computed on simple connected md2 graphs up to degree 10