ON SOMBOR ENERGY OF GRAPHS Текст научной статьи по специальности «Математика»

On Sombor energy of graphs

K. J. Gowtham, Narahari Narasimha Swamy

Dept. of Mathematics, University College of Science, Tumkur University, Tumakuru, Karnataka State, Pin 572 103, India

[email protected], [email protected]

DOI 10.17586/2220-8054-2021-12-4-411-417

The concept of Sombor index SO(G) was recently introduced by Gutman in the chemical graph theory. It is a vertex-degree-based topological

index and is denoted by SO(G). This paper introduces a new matrix for a graph G, called the Sombor matrix, and defines a new variant of

graph energy called Sombor energy ES(G) of a graph G. The striking feature of this new matrix is that it is related to well-known degree-based

topological indices called forgotten indices. When ES(G) values of some molecules containing hetero atoms are correlated with their total n-

electron energy, we got a good correlation with the correlation coefficient r = 0.976. Further, we found some bounds and characterizations on the

largest eigenvalue of S(G) and Sombor energy of graphs.

Keywords: Sombor index, Sombor energy, forgotten index.

Received: 30 May 2021

Revised: 1 July 2021

Final revision: 20 July 2021

1. Introduction

Spectral graph theory plays an important role in analyzing the matrices of graphs with the help of matrix theory and linear algebra. Now, spectral graph theory has attracted the attention of both pure and applied mathematicians whose benefit lies far from the spectral graph theory, which may be surprised because graph energy is a special kind of matrix norm. They will then recognize that the concept of graph energy (under different names) is encountered in several seemingly unrelated areas of their own expertise. The eigenvalues are closely related to almost all major invariants of a graph, linking one extremal property to another, they play a central role in the fundamental understanding of graphs [1]. In 1978, Gutman related the Graph energy and total n-electron energy in a molecular graph; it was defined as, the sum of absolute values of the eigenvalues of the associated adjacency matrix of a graph G [2]. Later, many matrices were defined based on distance and adjacency among the vertices, degree of the vertices involved in forming the graph structure like: Zagreb matrix [3], Randic matrix [4], distance matrix [5], Seidel matrix [6], Laplacian matrix [7], Seidel Laplacian matrix [8], signless Laplacian matrix [9], Seidel signless Laplacian matrix [10], degree sum matrix [11], etc.

Topological indices are mainly categorized into two types: namely degree-based indices and distance-based indices. Some of the well-known degree-based indices are first Zagreb index, second Zagreb index, forgotten index, hyper Zagreb index, Randic index, harmonic index, geometric-arithmetic index, redefined third Zagreb index, inverse sum index, etc. The details on degree-based topological indices we refer to [12-14]. Recently, Gutman et al. [15] defined new degree-based indices, called the Sombor index, which is one of the trending areas in the present graph-theoretical research. The details of this new index we refer to [16-25]. In [26], the chemical applicability of Sombor indices had been studied. The wide application of Sombor indices motivated us to define the Sombor matrix and Sombor energy of the graph.

2. Preliminaries

In this paper, we considered simple, finite, undirected, and connected graphs. A graph G involves a vertex set V = V(G) = jvi, v2,..., vn} and E = E(G) as its edge set. If two vertices have a common edge, then they are known as adjacent vertices. Likewise, if two edges have a common end vertex, then it is called an adjacent edge. The number of edges incident to a vertex v is called the degree of that vertex v and it is denoted by, dv The first and second Zagreb indices of a graph G [2] is defined as,

Mi(G)= ^ (du + dv )= ^ dU and M2(G) = ^ (d„dv) (1)

uveE(G) uev (G) uveE(G)

respectively.

Followed by the above definition, Furtula and Gutman introduced the forgotten topological index [27], defined as:

F (G) = £ (dU + ) = £ dU. (2)

uveE(G) ueV(G)

In [15] Gutman defined new degree based topological index called Sombor index, denoted by SO(G) and defined

as:

SO(G) = £ V(dU + dV), (3)

uveE(G)

In the next section, we introduce a new matrix for a graph G and a new graph invariant based on this matrix. 3. Sombor matrix and Sombor energy of graph

Definition 3.1. The Sombor matrix of a graph G with a vertex set V = V (G) = {v1,v2, v3,..., vn} and edge set E = E(G) is defined as S(G) = (sj)nxn, where:

^ = i^di+d,2, uv e E(G); j 10, otherwise;

where du denotes the degree of the vertex u.

The Sombor polynomial of a graph G is defined as:

Ps(G)(X) = IXI - S(G)|,

where I is an n x n unit matrix.

Since S(G) is real symmetric matrix, all roots of PS(G)(X) = 0 are real. Hence, they can be arranged as X1 > X2 > X3 ■ ■ ■ > Xn. The Sombor energy of graph G is given by

ES(G) = £ |Xi|.

4. Chemical applicability of ES(G)

The development of Huckel molecular orbital theory is mainly concentrated on conjugated, all carbon compounds. The range of those compounds can be studied if hetero atoms are considered. This can be done by comparing energy values for hetero compounds. To this end, we need to adjust Coulomb (a) and resonance integral (P) values for hetero atoms using the relations:

a! = a + hp and 0 = kft,

where h and k are correction values which are different and depending on what atom is in conjugation. So, we can take more than one value for a for a hetero atom but depends on the number of electrons hetero atom donates to n-system. Consider, the secular matrix of the compound urea:

' a - E 1.1310 0 0

1.1310 a +1.50 - E 0 0

0 0 a + 0 - E 0

0 0 0 a + 1 . 50 - E

By substituting appropriate values of a and 0 for urea in the above matrix and expanding the secular determinant, n-electron energy for urea can be calculated [28,29] we found the close resemblance between secular matrix of hetro molecule and Sombor matrix S(G) of corresponding molecular graph G. Further, we calculated ES(G) with dataset of total n-electron energy values of hetero atoms which are found in [30]. From Fig. 1 we found that ES(G) has good correlation hetero atoms, as mentioned in Fig. 2, with correlation coefficient r = 0.976 and r2 (adjusted) = 0.952.

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5. Some result on Sombor matrix

Theorem 5.1. Let G be graph with vertex set V(G) and edge set E(G) with the Sombor matrix S(G). If

Ps(G) = coXn + ciXn-i + C2Xn-2 + ■ ■ ■ + c„ is the characteristic polynomial of S(G), then

i. c2 = -F(G), _

ii. C3 = -2£ n VdU + dv,

A uvEe(A)

where A is a triangle in the graph G.

Proof. i. From the definition of PS(G), we have:

V^ „2 _ V^ jv , ,,2

cv = E

1<i<j<n

0 Sij Sji 0

ii. From the definition of PS(G), we have:

c3 = - E

1<i<j<n

0 sij sik sji 0 sjk ski skj 0

^ sj = Z, du + dV = -F (G).

1<i<j<n uvEE(G)

- E sij sjk ski = -2E II V7d2u + d2.

1<i<j<k<n A uvEE(A)

□

Lemma 5.2. [Newton's identity] Given an n x n matrix A, let P(X) = c0Xn + c1Xn-1 + c2Xn-2 + ■ ■ ■ + cn be the

characteristic polynomial of A, then the coefficient c3 = 1 [-(Tr(A))3 + 3 Tr(A) Tr(A2) - 2 Tr(A3)].

Theorem 5.3. Let S(G) = (sij )nxn be the Sombor matrix of a graph G and be its X1 > X2 > ■ ■ ■ > Xn eigenvalues, then:

n

i. £ X2 = 2F(G), i=1

n

ii. £ X3 =3 £ n Vdu + dV,

i=1 A uveE(A)

where A is a triangle in the graph G

Proof.

n n n

2x2 = Tr(S(G)2) = ££ sij sji = £ s2i + £ sij sji = 2 £ s2 = 2 £ du + dV

= 1 i=1 j = 1 i=1 i=j i<j uveE(G)

ii. We know that, Tr(S(G)3) = £ X3. The result follows by equating the coefficient c3 values given in The-

i=1

orem 5.1, Lemma 5.2 and using the facts Tr(S(G)) = 0 and Tr(S(G)2) = 2F(G). We get the required result.

□

6. Bounds for the largest eigenvalue

Theorem 6.1. If G is any graph with n verties with S(G) = (sij )nxn being its Sombor matrix and X1 > X2 > ■ ■ ■ > Xn are its eigenvalues, then

X1 < ^2(n - 1)F(G). Proof. Taking ai = Xi and bi = 1 for i = 1,2,3,... ,n in Cauchy-Schwarz inequality we get:

(n \ 2 n

£ Xj < (n -1) £ x2.

i=2 i=2

On solving we get:

(-X1)2 < (n - 1)(2F(G) - X2) X1 < V2(n - 1)F(G).

7. Bounds for Sombor energy

Now we obtain some bounds on the Sombor energy of graphs. To this end, we make use of the following classical inequalities

Lemma 7.1. [Diaz-Metcalf Inequality] Let (a1, a2,..., an) and (b1,b2,..., bn) be positive real numbers, satisfying the condition rai < bi < Rai for 1 < i < n. Then:

n n n

£ b2 + rR£ a2 < (r + R) £ aibi.

i=1 i=1 i=1

Equality holds if and only if bi = Rai or bi = rai for 1 < i < n. Lemma 7.2. Let a1 ,a2,... ,an be non-negative real numbers, then:

1 ^ V/n^ ^ A2 , 1 ^ \1/n

ninAai-U±ai) <ni^ai- v^i) <n(n- 1HnAai-UJ-N i=1 \i=1 ) / i=1 \i=1 ) \ i=1 \i=1 )

7.1. Lower bounds for the Sombor energy

Theorem 7.3. Let G be any graph with n verties and let P be the absolute value of the determinant of Sombor matrix S(G), then:

^2F(G) + n(n - 1)P2/n < ES(G).

Proof.

( n \ 2 n

[ES(G)]2 = £ |XiM = £ |XiI2 + £ |Xi||XjI = 2F(G) + £ |Xi||Xj|.

\i=1 / i=1 i=j i=j

Clearly we have:

( 2/n 2

1 ^ T-r ,,, ,,, ,a/(n(n-1))

( |Xi||Xj(|Xi||X.

n(n 1)

i=j i=j

Xi

i=1

Pn E|Xi||Xj|> n(n - 1)P2/n,

i=j

therefore

^2F(G) + n(n - 1)P2/n < ES(G).

□

Theorem 7.4. Let G be a graph with n verties. Then

2F (G) + n|X1||Xn| |Xn|+|X1I

< ES(G),

where, |X1| and |Xn| are maximum and minimum of the absolute value of eigenvalues of S(G). Equality will be attained if and only if for each 1 < i < n, either |Xi| = |X1| or |Xi| = |Xn|.

Proof. substituting bi = |Xi|, ai = 1, r = |Xn| and R = |X1| in lemma 7.1, we have

nn

£ |Xi|2 + |Xn||X1| £ 1 < (|X1| + IX1I) ES(G) < ES(G).

I2

= 1 i=1 2F (G) + nIX1IIXnI

IX11 + IXnI

7.2. Upper bound for Sombor index Theorem 7.5. If G is a graph with n verties, then

ES(G) < VnF(G). Proof. Put ai = 1 and bi = I Xi I in Cauchy-Schwarz inequality, we get

n2

2

[ES(G)]2 < n^ IXiIj = nF(G).

Simplifying the above equation we get the required result. □

Theorem 7.6. If G is a graph with n verties, then:

ES(G) < n|2F(G) + [Det(N(G)2)]1/n| - 2F(G),

where I Det(N (G)) I is absolute value of the determinant ofSombor matrix S (G). Proof. Substituting ai = X2 for i = 1,2,... ,n in lemma7.2, we have:

( n ( n \ 1/n\ n ( n \ 2

n £ X2 - n X2 I < n £ X? - £ Xi) .

i=1 \i=1 / j i=1 \i=1 /

Using results in theorem 5.1, we get:

n ^~2F(G) - I Det(S(G))I1/^ < n2F(G) - (ES(G))2 .

On simplifying above equation we arrive the required result. □

8. Conclusion

Recently, Gutman introduced a new vertex-degree-based topological index, called the Sombor index SO(G) in chemical graph theory. In this paper, we have introduced a new matrix for a graph G, called the Sombor matrix, and defined a new variant of graph energy called the Sombor energy ES(G) of a graph G. The striking feature of this new matrix is that it is related to the well-known degree-based topological indices called forgotten indices. When ES(G) values of some molecules containing hetero atoms are correlated with their total n-electron energy, we obtained a good correlation with the correlation coefficient r = 0.952. Further, bounds (lower and upper) and characterizations on the largest eigenvalue of S(G) and Sombor energy of graphs have been studied.

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