TY - JOUR

T1 - On Some Topological Indices Defined via the Modified Sombor Matrix

AU - Zuo, Xuewu

AU - Rather, Bilal Ahmad

AU - Imran, Muhammad

AU - Ali, Akbar

N1 - Funding Information:
This research work is supported by the Natural Science Foundation of Anhui Province Higher School (No.KJ2020A0780), China, and the United Arab Emirates University (UAEU) via Grant No. G00003739.
Publisher Copyright:
© 2022 by the authors.

PY - 2022/10

Y1 - 2022/10

N2 - Let G be a simple graph with the vertex set (Formula presented.) and denote by (Formula presented.) the degree of the vertex (Formula presented.). The modified Sombor index of G is the addition of the numbers (Formula presented.) over all of the edges (Formula presented.) of G. The modified Sombor matrix (Formula presented.) of G is the n by n matrix such that its (Formula presented.) -entry is equal to (Formula presented.) when (Formula presented.) and (Formula presented.) are adjacent and 0 otherwise. The modified Sombor spectral radius of G is the largest number among all of the eigenvalues of (Formula presented.). The sum of the absolute eigenvalues of (Formula presented.) is known as the modified Sombor energy of G. Two graphs with the same modified Sombor energy are referred to as modified Sombor equienergetic graphs. In this article, several bounds for the modified Sombor index, the modified Sombor spectral radius, and the modified Sombor energy are found, and the corresponding extremal graphs are characterized. By using computer programs (Mathematica and AutographiX), it is found that there exists only one pair of the modified Sombor equienergetic chemical graphs of an order of at most seven. It is proven that the modified Sombor energy of every regular, complete multipartite graph is (Formula presented.) ; this result gives a large class of the modified Sombor equienergetic graphs. The (linear, logarithmic, and quadratic) regression analyses of the modified Sombor index and the modified Sombor energy together with their classical versions are also performed for the boiling points of the chemical graphs of an order of at most seven.

AB - Let G be a simple graph with the vertex set (Formula presented.) and denote by (Formula presented.) the degree of the vertex (Formula presented.). The modified Sombor index of G is the addition of the numbers (Formula presented.) over all of the edges (Formula presented.) of G. The modified Sombor matrix (Formula presented.) of G is the n by n matrix such that its (Formula presented.) -entry is equal to (Formula presented.) when (Formula presented.) and (Formula presented.) are adjacent and 0 otherwise. The modified Sombor spectral radius of G is the largest number among all of the eigenvalues of (Formula presented.). The sum of the absolute eigenvalues of (Formula presented.) is known as the modified Sombor energy of G. Two graphs with the same modified Sombor energy are referred to as modified Sombor equienergetic graphs. In this article, several bounds for the modified Sombor index, the modified Sombor spectral radius, and the modified Sombor energy are found, and the corresponding extremal graphs are characterized. By using computer programs (Mathematica and AutographiX), it is found that there exists only one pair of the modified Sombor equienergetic chemical graphs of an order of at most seven. It is proven that the modified Sombor energy of every regular, complete multipartite graph is (Formula presented.) ; this result gives a large class of the modified Sombor equienergetic graphs. The (linear, logarithmic, and quadratic) regression analyses of the modified Sombor index and the modified Sombor energy together with their classical versions are also performed for the boiling points of the chemical graphs of an order of at most seven.

KW - Sombor index

KW - adjacency matrix

KW - correlation

KW - modified Sombor energy

KW - modified Sombor matrix

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U2 - 10.3390/molecules27196772

DO - 10.3390/molecules27196772

M3 - Article

C2 - 36235303

AN - SCOPUS:85139886859

SN - 1420-3049

VL - 27

JO - Molecules

JF - Molecules

IS - 19

M1 - 6772

ER -