Some Graft Transformations and Their Applications on $Q$-Spectral Radius of a Graph

Abstract

A generalized $\theta$-graph is composed of at least three internal disjoint paths (at most one of them is with length 1) which have the same initial vertex and the same terminal vertex. If the initial vertex and the terminal vertex are the same in a generalized $\theta$-graph, then the generalized $\theta$-graph is called a degenerated $\theta$-graph or a petal graph. In this paper, two graft transformations that increase or decrease the $Q$-spectral radius of a graph are represented. With them, for the generalized $\theta$-graphs and petal graphs with order $n$, the extremal graphs with the maximal $Q$-spectral radius and the extremal graphs with the minimal $Q$-spectral radius are characterized, respectively.