Extremal results on the spectral radius of function-weighted adjacency matrices

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 $C^2$-function in ${\left(R^{+}\right)}^2$ and be symmetric in $x$ and $y$. If for every $\bar{x}$ such that $\left\{\bar{x}\right\}\times R^{+}\ne \varnothing$, then $f_x^{\prime }(\bar{x},y)\ge 0$ (resp. $f_x^{\prime \prime }(\bar{x},y)\ge 0$ ) for all $(\bar{x},y)\in {\left(R^{+}\right)}^2$, we say $f(x,y)$ is increasing (resp. convex) in variable $x$. The function-weighted adjacency matrix $A_f\left(G\right)$ 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.

In this paper, we consider the unimodality of the principal eigenvector of the path $P_n$ and characterize the tree on $n$ vertices with the smallest $A_f$-spectral radius of $G$ under the condition that $f(x,y)>0$ is increasing and convex in variable $x$. We also obtain the unicyclic graph on $n$ vertices with the largest $A_f$-spectral radius of $G$ under the same condition.