Maximum Detour Index of Unicyclic Graphs with Given Maximum Degree

Abstract

The detour index of a connected graph is defined as the sum of detour distances between all its unordered vertex pairs. We determine the maximum detour index of $n$-vertex unicyclic graphs with maximum degree $\mathrm{\Delta }$, and characterize the unique extremal graph, where $2\le \mathrm{\Delta }\le n-1$.