The maximum degree resistance distance of cacti

Abstract

Various topological indices, based on the distances between the vertices of a graph, are widely used in theoretical chemistry. The degree resistance distance of a graph $G$ is defined as ${D_R}(G) = \sum\limits_{\{u,v\} \subseteq V(G)} {[d(u) + d(v)]R(u,v)},$ where $d(u)$ is the degree of the vertex $u,$ and $R(u, v)$ the resistance distance between the vertices $u$ and $v.$ A graph $G$ is called a cactus if each block of $G$ is either an edge or a cycle. In this paper, we completely characterize the extremal cacti having the maximum degree resistance distance among all cacti with $n$ vertices and $t$ cycles, and extend some results of a recent paper [J. Tu, J. Du, G. Su, The unicyclic graphs with maximum degree resistance distance, Appl. Math. Comput. 268 (2015) 859-864].