Extremal trees with respect to spectral radius of restrictedly weighted adjacency matrices

Abstract

For a graph $G=(V,E)$ and $v_i\in V$, denote by $d_i$ the degree of vertex $v_i$. Let $f(x,y)>0$ be a real symmetric function in $x$ and $y$. The weighted adjacency matrix $A_f(G)$ of a graph $G$ is a square matrix, where the $(i,j)$-entry is equal to $f(d_i,d_j)$ if the vertices $v_i$ and $v_j$ are adjacent and 0 otherwise. Li and Wang \cite{U9} tried to unify methods to study spectral radius of weighted adjacency matrices of graphs weighted by various topological indices. If $f_x^{\prime }(x,y)\ge 0$ and $f_x^{\prime \prime }(x,y)\ge 0$, then $f(x,y)$ is said to be increasing and convex in variable $x$, respectively. They obtained the tree with the largest spectral radius of $A_f(G)$ is a star or a double star when $f(x,y)$ is increasing and convex in variable $x$. In this paper, we add the following restriction: $f(x_1,y_1)\ge f(x_2,y_2)$ if $x_1+y_1=x_2+y_2$ and $\mid x_1-y_1\mid >\mid x_2-y_2\mid$ and call $A_f(G)$ the restrictedly weighted adjacency matrix of $G$. The restrictedly weighted adjacency matrix contains weighted adjacency matrices weighted by first Zagreb index, first hyper-Zagreb index, general sum-connectivity index, forgotten index, Somber index, $p$-Sombor index and so on. We obtain the extremal trees with the smallest and the largest spectral radius of $A_f(G)$. Our results push ahead Li and Wang's research on unified approaches.