On spectrum of Sombor matrix and Sombor energy of graphs
Abstract
Abstract The Sombor index ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>SO</m:mi> </m:math> {\mathrm{SO}} ) is a recently introduced degree-based graph invariant, defined as the sum over all pairs of adjacent vertices <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mi>v</m:mi> </m:mrow> </m:math> {u,v} of the term <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msqrt> <m:mrow> <m:msubsup> <m:mi>d</m:mi> <m:mi>u</m:mi> <m:mn>2</m:mn> </m:msubsup> <m:mo>+</m:mo> <m:msubsup> <m:mi>d</m:mi> <m:mi>v</m:mi> <m:mn>2</m:mn> </m:msubsup> </m:mrow> </m:msqrt> </m:math> {\sqrt{{d_{u}^{2}+d_{v}^{2}}}} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>d</m:mi> <m:mi>u</m:mi> </m:msub> </m:math> {d_{u}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>d</m:mi> <m:mi>v</m:mi> </m:msub> </m:math> {d_{v}} denote the degrees of vertices u and v , respectively. The matrix associated with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>SO</m:mi> </m:math> {\mathrm{SO}} is the Sombor matrix, and its spectrum is the Sombor spectrum. In this paper, the connected graphs having exactly two and exactly three Sombor eigenvalues are characterized. Bounds are obtained for the spectral radius and energy of the Sombor matrix, and the corresponding extremal graphs are determined. In addition, the Sombor spectra of several families of graphs are calculated.