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Abstract
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.
Highlights
This paper is concerned with simple graphs, i.e. with graphs without multiple, directed, or weighted edges, and without loops
We prove the following: Theorem 1
( 1) p(s) 2q(s) w(s) is: positive if s contains no cycles of size divisible by 4, negative if s contains an odd number of cycles of a size divisible by 4, and positive if s contains an even number of cycles of a size divisible by 4
Summary
This paper is concerned with simple graphs, i.e. with graphs without multiple, directed, or weighted edges, and without loops. In the mathematical (Cruz et al, 2015), (Das et al, 2018), (Furtula et al, 2013), (Liu et al, 2019), (Rada & Cruz, 2014), (Zhong & Xu, 2014) and chemical (Todeschini & Consonni, 2009) literature, several dozens of vertex-degree-based graph invariants (usually referred to as “topological indices”) have been introduced and extensively studied. Their general formula is VDBI VDBI (G). In order to prove Theorem 1, we need some preparations
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