Abstract:

Let $G_\omega=(G, \omega)$ be a weighted graph, whose adjacency matrix and weighted degree diagnoal matrix are $A(G_\omega)$ and $D(G_\omega)$, respectively. For given $\alpha \in [0, 1]$, the matrix $A_\alpha(G_\omega)=\alpha D(G_\omega)+(1-\alpha)A(G_\omega)$ is the $A_\alpha$- matrix of $G_\omega$. In this paper, we give some bounds on the $A_\alpha$-eigenvalue of connected weighted non-regular graphs $G_\omega$, and obtain the lower bound of the ratio of the largest component to the smallest component in the eigenvector of the $A_\alpha$- spectral radius.

Key words: weighted graph, A_α-matrix, A_α-spectral radius