Extremal trees and unicyclic graphs with respect to spectral radius of weighted adjacency matrices with property $P^{*}$

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 $\displaystyle f(d_{i}, d_{j})$ if the vertices $v_{i}$ and $v_{j}$ are adjacent and 0 otherwise. Li and Wang (Linear Algebra Appl 620:61–75, 2021) tried to unify methods to study spectral radius of the weighted adjacency matrices of graphs weighted by various topological indices. If $\displaystyle f'_{x}(x, y)\ge 0$ and $\displaystyle f''_{x}(x, y)\ge 0$, then $\displaystyle f(x, y)$ 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. If for any $x_{1}+y_{1}=x_{2}+y_{2}$ and $\mid x_{1}-y_{1}\mid >\mid x_{2}-y_{2}\mid$, $f(x_{1},y_{1})>f(x_{2},y_{2})$, then $\displaystyle f(x, y)$ is said to have the property P. In this paper, if $f(x, y)>0$ is increasing and convex in variable x and for any $x_{1}+y_{1}=x_{2}+y_{2}$ and $\mid x_{1}-y_{1}\mid >\mid x_{2}-y_{2}\mid$, $f(x_{1},y_{1})\ge f(x_{2},y_{2})$, then we say $\displaystyle f(x, y)$ has the property $P^{*}$ and call $A_{f}(G)$ the weighted adjacency matrix with property $P^{*}$ of G. It contains the 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 largest spectral radii, and the extremal unicyclic graphs with the smallest and first three maximum spectral radii of $A_{f}(G)$. Our results push ahead Li and Wang’s research on unified approaches.