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Extremal trees with respect to spectral radius of restrictedly weighted adjacency matrices | | Ruiling Zheng
; Xiaxia Guan
; Xian an Jin
; | | Date: |
5 Dec 2022 | | 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
cite{U9} tried to unify methods to study spectral radius of weighted adjacency
matrices of graphs weighted by various topological indices. If $displaystyle
f’_{x}(x, y)geq0$ and $displaystyle f’’_{x}(x, y)geq0$, 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 $x$. In
this paper, we add the following restriction: $f(x_{1},y_{1})geq
f(x_{2},y_{2})$ 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. | | Source: | arXiv, 2212.02247 | | Services: | Forum | Review | PDF | Favorites | |
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