On the $\alpha$-index of graphs with pendent paths

Abstract

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For every real $\alpha \in \left[0,1\right]$, write $A_{\alpha }\left(G\right)$ for the matrix \[ A_α\left( G\right) =αD\left( G\right) +(1-α)A\left( G\right) . \] This paper presents some extremal results about the spectral radius ${\rho }_{\alpha }\left(G\right)$ of $A_{\alpha }\left(G\right)$ that generalize previous results about ${\rho }_0\left(G\right)$ and ${\rho }_{1/2}\left(G\right)$. In particular, write $B_{p,q,r}$ be the graph obtained from a complete graph $K_p$ by deleting an edge and attaching paths $P_q$ and $P_r$ to its ends. It is shown that if $\alpha \in \left[0,1\right)$ and $G$ is a graph of order $n$ and diameter at least $k,$ then% \[ ρ_α(G)\leqρ_α(B_{n-k+2,\lfloor k/2\rfloor,\lceil k/2\rceil}), \] with equality holding if and only if $G=B_{n-k+2,\lfloor k/2\rfloor ,\lceil k/2\rceil }$. This result generalizes results of Hansen and Stevanović \cite{HaSt08}, and Liu and Lu \cite{LiLu14}.