Characterizing tricyclic graphs with pendant vertices having largest $A_{\hat{I}}\pm$-spectral radius

Abstract

For a graph $G$ with adjacency matrix $A(G)$ and degree diagonal matrix $D(G)$, the $A_{\hat{I}}\pm$-matrix of $G$ is defined as \begin{equation*} A_α(G) = αD(G) + (1- α) A(G), \text{ for any } α\in [0,1]. \end{equation*} The $A_{\hat{I}}\pm$-spectral radius of $G$ is the largest eigenvalue of the matrix $A_{\hat{I}}\pm (G)$. A tricyclic graph of order $n$ is a simple connected graph with $n+2$ edges. In this paper, we characterize the unique graph having the largest $A_{\hat{I}}\pm$-spectral radius for $\hat{I}\pm \in [\frac{1}{2},1)$ among all tricyclic graphs of order $n$ with $k(\ge 1)$ pendant vertices. As an application, we derive a sufficient spectral condition (alternate to the edge condition) to guarantee the absence of the tricyclic structure in a graph with $k$ pendant vertices.