RELATING GRAPH ENERGY WITH VERTEX-DEGREE-BASED ENERGIES

Introduction/purpose: The paper presents numerous vertex-degree-based graph invariants considered in the literature. A matrix can be associated to each of these invariants. By means of these matrices, the respective vertex-degree-based graph energies are defined as the sum of the absolute values of the eigenvalues. Results: The article determines the conditions under which the considered graph energies are greater or smaller than the ordinary graph energy (based on the adjacency matrix). Conclusion: The results of the paper contribute to the theory of graph energies as well as to the theory of vertex-degree-based graph invariants.


RELATING GRAPH ENERGY WITH VERTEX-DEGREE-BASED ENERGIES Introduction
This paper is concerned with simple graphs, i.e. with graphs without multiple, directed, or weighted edges, and without loops. Let G be such a graph with n vertices, labeled as 1 2 , , , n v v v  . Two vertices connected by an edge are said to be adjacent. The degree of the vertex i v , denoted by deg( ) i v , is the number of the first neighbors of i v .
For each function ( , ) F x y and each graph G, a symmetric square The respective vertex-degree-based graph energy (of the graph G) is equal to the sum of absolute values of the eigenvalues of For some of the above given functions ( , ) Such are the functions pertaining to the Randić, harmonic, sum-connectivity, and geometricarithmetic indices. For some of the above given functions, Such are those related to the first and second Zagreb, extended, forgotten, and arithmetic-geometric indices, as well as for some reciprocal and inverse indices. For such functions, we prove the following: The equality cases will be considered later.
In order to prove Theorem 1, we need some preparations.

Preliminary considerations
be a polynomial with all zeros real. Then its energy satisfies (Mateljević et al, 2010) As well known, a graph is bipartite if and only if it does not contain cycles of odd size. The characteristic polynomial of a bipartite graph is of the form The respective energies are then Proving Theorem 1 We apply the Sachs coefficient theorem (Cvetković et al, 2010), (Gutman, 2017a). Recall that a Sachs graph is a graph consisting of vertices of degree one and/or two, i.e., all its components are isolated edges and/or cycles.
The application of the Sachs theorem to the coefficients of ( , ) F G x In view of the above, the contribution of the Sachs graph s to the term There remains a case when i i   is an odd integer. Then, s has at least one cycle whose size is divisible by 4. Let, for the sake of simplicity, this be a single 12-membered cycle, whose edges are which is non-negative. Because of ( ) 1 w e  , this term is also less than or equal to unity. Thus, also in this case, the joint contribution of the Sachs graphs , ', '' s s s to ( ) F E G is positive but not greater than their contribution to ( ) E G .
This completes the proof of Theorem 1(a).
The proof of Theorem 1(b) is analogous. Note that the special case of Theorem 1(b), pertaining to extended energy, was earlier communicated in (Gutman, 2017b).

Discussion
If the graph G is not bipartite, then it contains odd cycles. Then, of course, some Sachs graphs also contain odd cycles. Consequently, some of the coefficients 2 1 For these reasons, it is not easy to extend Theorem 1 to non-bipartite graphs, and we leave this for some later moment or some more skilled colleague. Randić and harmonic energies, etc. Interestingly but evidently, the Albertson and sigma energies of regular graphs are equal to zero. On the other hand, for the class of stepwise irregular graphs (Gutman, 2018), the Albertson and sigma matrices coincide with the adjacency matrix, and for such graphs the Albertson and sigma energies are equal to the ordinary graph energy.