The Spectral Properties of $p$-Sombor (Laplacian) Matrix of Graphs

Abstract:

      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 index 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 index matrix $(p=-1)$.