On the $A_{\alpha }$-spectral radius of graphs without linear forests

Abstract

Let $A(G)$ and $D(G)$ be the adjacency and degree matrices of a simple graph $G$ on $n$ vertices, respectively. The \emph{$A_{\alpha }$-spectral radius} of $G$ is the largest eigenvalue of $A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G)$ for a real number $\alpha \in [0,1]$. In this paper, for $\alpha \in (0,1)$, we obtain a sharp upper bound for the $A_{\alpha }$-spectral radius of graphs on $n$ vertices without a subgraph isomorphic to a liner forest for $n$ large enough and characterize all graphs which attain the upper bound. As a result, we completely obtain the maximum signless Laplacian spectral radius of graphs on $n$ vertices without a subgraph isomorphic to a liner forest for $n$ large enough.

18 pages