Characterizing tricyclic graphs with pendant vertices having largest $A_α$-spectral radius

Abstract

For a graph $G$ with adjacency matrix $A(G)$ and degree diagonal matrix $D(G)$, the $A_α$-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 $A_α$-spectral radius of $G$ is the largest eigenvalue of the matrix $A_α(G)$. 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 $A_α$-spectral radius for $α\in [\frac{1}{2}, 1)$ among all tricyclic graphs of order $n$ with $k (\geq 1)$ 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.

Figures (9)

• Figure 1: The graph $\mathcal{T}_3$.
• Figure 2: All possible configurations of cycles in graphs of class $\mathscr{T}_n^k$.
• Figure 3: Graphs $G_1$, $G_3$ and $G_3$.
• Figure 4: Only possible cycle configuration in graphs of $\mathscr{T}_n^{k,7}$.
• Figure 5: All possible configurations of six cycles in graphs of $\mathscr{T}_n^{k,6}$.

Theorems & Definitions (32)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• proof
• Lemma 2.7