Extremal spectral radius of weighted adjacency matrices of bicyclic graphs

Abstract

The weighted adjacency matrix $A_f(G)$ of a simple graph $G=(V,E)$ is the $|V|\times |V|$ matrix whose $ij$-entry equals $f(d_i,d_j)$, where $f(x,y)$ is a symmetric function such that $f(d_i,d_j)>0$ if $ij\in E$ and $f(d_i,d_j)=0$ if $ij\notin E$ and $d_i$ is the degree of the vertex $i$. In this paper, we determine the unique graph having the largest spectral radius of $A_f(G)$ among all the bicyclic graphs under the assumption that $f(x,y)$ is increasing and convex in $x$ and $f(x_1,y_1)\ge f(x_2,y_2)$ when $|x_1-y_1|>|x_2-y_2|$ and $x_1+y_1=x_2+y_2$. Moreover, we determine the unique graph having the second largest spectral radius of $A_f(G)$ among all the bicyclic graphs when $f(x,y)=x+y$, $(x+y{)}^2$ or $x^2+y^2$, which corresponds to the well-known first Zagreb index, first hyper-Zagreb index, and forgotten index, respectively. In addition, we also characterize the bicyclic graphs with the first two largest spectral radii of $A_f(G)$ when $f(x,y)=\frac{1}{2}(x/y+y/x)$, corresponding to the extended index.