On the Index of Quasi-Tree Graphs with Perfect Matchings

Abstract

A connected graph \(G = (V(G), E(G))\) is called a quasi-tree graph if there exists a vertex \(u_0 \in V(G)\) such that \(G – u_0\) is a tree. Let \(\mathcal{P}(2k) := \{G: G \text{ is a quasi-tree graph on } 2k \text{ vertices with perfect matching}\}\), and \(\mathcal{P}(2k, d_0) := \{G: G \in \mathcal{P}(2k), \text{ and there is a vertex } u_0 \in V(G) \text{ such that } G – u_0 \text{ is a tree with } d_G(u_0) = d_0\}\). In this paper, the maximal indices of all graphs in the sets \(\mathcal{P}(2k)\) and \(\mathcal{P}(2k, d_0)\) are determined, respectively. The corresponding extremal graphs are also characterized.