SPECTRUM AND ENERGY OF THE SOMBOR MATRIX

Abstract: Introduction/purpose: The Sombor matrix is a vertex-degree-based matrix associated with the Sombor index. The paper is concerned with the spectral properties of the Sombor matrix. Results: Equalities and inequalities for the eigenvalues of the Sombor matrix are obtained, from which two fundamental bounds for the Sombor energy (= energy of the Sombor matrix) are established. These bounds depend on the Sombor index and on the „forgotten“ topological index. Conclusion: The results of the paper contribute to the spectral theory of the Sombor matrix, as well as to the general spectral theory of matrices associated with vertex-degree-based graph invariants.


Introduction
In this paper, we are concerned with simple graphs, i.e., graphs without directed, weighted, or multiple edges, and without self-loops. Let  For other graph-theoretical notions, the readers are referred to standard textbooks (Harary, 1969), (Bondy & Murthi, 1975).
In the mathematical and chemical literature, degree-based graph invariants of the form have been and are currently studied, where ( , ) f x y is an appropriately chosen function with the property ( , ) ( , ) f x y f y x  . The oldest such invariants were put forward as early as in the 1970s, and by now their number exceeds several dozens (Kulli, 2020), (Todeschini & Consonni, 2009 ( , ) f x y x y   (Furtula & Gutman, 2015), and the Sombor index for which 2 2 ( , ) f x y x y   (Gutman, 2021). Thus, these indices are defined as Recall that F is often written in the form which, of course, is equivalent to (2). The first paper on the Sombor index was published only a few months ago (Gutman, 2021). Because this graph invariant is based on using Euclidean metrics, it promptly attracted the attention of quite a few colleagues. As a consequence of this, numerous papers on Sombor index have already been published (Alikhani & Ghanbari, 2021), (Cruz & Rada, 2021), (Došlić et al, 2021), (Horoldagva & Xu, 2021), (Kulli, 2021), and more will appear in the near future. Bearing this in mind, we were motivated to investigate the matrix constructed from the Sombor index in an earlier proposed manner (Das et al, 2018), and to study some of its spectral properties.
Let the vertices of the graph G be labeled by 1,2,…,n. Then the (0,1)-adjacency matrix of G, denoted by ( ) A G , is defined as the symmetric square matrix of order n, whose (I,j)-element is The eigenvalues of ( ) A G form the spectrum of the graph G. For the details of the spectral graph theory see (Cvetković et al, 2010). Some time ago (Das et al, 2018), it was attempted to combine the spectral graph theory with the theory of vertex-degree-based graph invariants. For this, using formula (1), an adjacency-matrix-type square symmetric matrix The theory based on the matrix ( ) F A G and its spectrum was recently elaborated in some detail (Li & Wang, 2021), (Shao et al, 2021).
In this paper, we examine a special case of ( ) The eigenvalues of ( ) SO A G are denoted by 1 2 , ,..., n    , and are said to form the Sombor spectrum of the graph G. Then, as usual, the Sombor characteristic polynomial is defined as in analogy to the ordinary characteristic polynomial (Cvetković et al, 2010) Spectral properties of the Sombor matrix Proof. The first equality is a direct consequence of The second equality is obtained from (4) as follows. Suppose that the vertices of G are labeled by 1,2,…,n. Then, This completes the proof of Lemma 1.
Recalling that the sum of squares of the eigenvalues of the ordinary adjacency matrix is equal to 2m, from Lemma 1 we realize that in the spectral theory of the Sombor matrix, the forgotten topological index plays an analogous role as the number of edges plays in the ordinary spectral graph theory. This will be seen from the bounds for the Sombor energy, deduced in the forthcoming section (Theorems 1, 2, and 3). Proof. According to the Rayleigh-Ritz variational principle, if  is any n-dimensional column-vector, then

Setting
(1,1,...,1) T   , we get In the case of regular graphs, (1,1,...,1) T   is an eigenvector of ( ) SO A G , corresponding to the eigenvalue 1  . To see this, note that if G is a regular graph of a degree r, then (1,1,...,1) T   is an eigenvector of ( ) A G is a well-known fact (Cvetković et al, 2010). Then, and only then, equality in Lemma 2 holds.
By this, the proof of Lemma 2 has been completed. Proof. Start with the inequality whose validity is obvious:

Sombor energy and its bounds
which is the analogue of the Koolen-Moulton bound (Koolen & Moulton, 2001), namely Proof. We follow the reasoning from the paper (Koolen & Moulton, 2001), modified for the Sombor energy. In an analogous way as in the proof of Theorem 1, our starting point is This yields 2 ( ) | | ( 1) 2 ( ) Therefore, inequality (6) remains valid if on the right-hand side of ( ) x  , the variable x is replaced by the lover bound for i  , from Lemma 2. This results in Theorem 2.
Theorem 3. Let G be a bipartite graph on n vertices, with Sombor and forgotten indices ( ) SO G and ( ) F G , respectively. Then which, again is analogous to another Koolen-Moulton bound (Koolen & Moulton, 2003): Proof. Theorem 3, valid for bipartite graphs, is deduced in an analogous manner as Theorem 2, by starting with