The spectral study of p-Sombor (Laplacian) matrix of graphs
The spectral properties of p-Sombor (Laplacian) matrix of graphs
Received:May 20, 2022  Revised:November 01, 2022
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 Author Name Affiliation Address Hechao Liu School of Mathematical Sciences, South China Normal University School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, Lihua You South China Normal University
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The Sombor index, which was recently introduced into chemical graph theory, can predict physico-chemical properties of molecules. In this paper, we investigate the properties of ($p$-)Sombor index from an algebraic viewpoint. The $p$-Sombor matrix $\mathcal{S}_{p}(G)$ is the square matrix of order $n$ whose $(i,j)$-entry is equal to $((d_{i})^{p}+(d_{j})^{p})^{\frac{1}{p}}$ if $v_{i}\sim v_{j}$, and 0 otherwise, where $d_{i}$ denotes the degree of vertex $v_{i}$ in $G$. The matrix generalizes the famous Zagreb matrix $(p=1)$, Sombor matrix $(p=2)$ and inverse sum indeg matrix $(p=-1)$. In this paper, we find a pair of p-Sombor noncospectral equienergetic graphs and determine some bounds for the p-Sombor (Laplacian) spectral radius. Then we describe the properties of connected graphs with $k$ distinct p-Sombor Laplacian eigenvalues. At last, we determine the Sombor spectrum of some special graphs. As a by-product, we determine the spectral properties of Sombor matrix $(p=2)$, Zagreb matrix $(p=1)$ and inverse sum indeg matrix $(p=-1)$.