Abstract
Spectral graph theory plays an important role in engineering. Let G be a simple graph of order n with vertex set V=v1,v2,…,vn. For vi∈V, the degree of the vertex vi, denoted by di, is the number of the vertices adjacent to vi. The arithmetic-geometric adjacency matrix AagG of G is defined as the n×n matrix whose i,j entry is equal to di+dj/2didj if the vertices vi and vj are adjacent and 0 otherwise. The arithmetic-geometric spectral radius and arithmetic-geometric energy of G are the spectral radius and energy of its arithmetic-geometric adjacency matrix, respectively. In this paper, some new upper bounds on arithmetic-geometric energy are obtained. In addition, we present the Nordhaus–Gaddum-type relations for arithmetic-geometric spectral radius and arithmetic-geometric energy and characterize corresponding extremal graphs.
Highlights
Spectral properties of graphs have been widely applied in the field of engineering technology
Complex networks are often used as abstract models to study and reveal the relationship among system elements. e mathematical study of complex network depends on graph theory
We further study the arithmeticgeometric spectral radius and arithmetic-geometric energy
Summary
Spectral properties of graphs have been widely applied in the field of engineering technology. The development of systems engineering depends on the results and conclusions of basic studies in graph spectral theory. E combinatorial forms of spectral of weighted graphs, based on topological indices, are closely related to the molecular orbital energy levels of π-electrons in conjugated hydrocarbons, as well as the spectrum properties. We further study the arithmeticgeometric spectral radius and arithmetic-geometric energy. Some bounds on AG spectral radius and AG energy were obtained in a couple of papers [4, 5]. E rest of the paper is structured as follows: in Section 2, we give some useful lemmas; in Section 3, we get some new bounds on the AG energy; and in Sections 4 and 5, we obtain the Nordhaus–Gaddum-type relations for the AG spectral radius and AG energy of graph G, respectively For other undefined notions and terminologies from graph theory, the readers are referred to [20, 21]. e rest of the paper is structured as follows: in Section 2, we give some useful lemmas; in Section 3, we get some new bounds on the AG energy; and in Sections 4 and 5, we obtain the Nordhaus–Gaddum-type relations for the AG spectral radius and AG energy of graph G, respectively
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