Let [Formula: see text] be a simple undirected graph with vertex set [Formula: see text]. The arithmetic–geometric matrix [Formula: see text] of a graph [Formula: see text] is defined so that its [Formula: see text]-entry is equal to [Formula: see text] if the vertices [Formula: see text] and [Formula: see text] are adjacent, and zero otherwise, where [Formula: see text] denotes the degree of vertex [Formula: see text] in [Formula: see text]. In this paper, some bounds on the arithmetic–geometric spectral radius and arithmetic–geometric energy are obtained, and the respective extremal graphs are characterized. Moreover, some bounds for the arithmetic–geometric Estrada index involving arithmetic–geometric energy of graphs are determined. Finally, a class of arithmetic–geometric equienergetic graphs is constructed by graph operations.

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Let $G$ be a simple undirected graph. For any real number $\alpha \in[0,1]$, Nikiforov defined the $A_{\alpha}$-matrix of $G$ as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the degree diagonal matrix of $G$, respectively. The $A_{\alpha}$-spread of a graph is defined as the difference between the largest eigenvalue and the smallest eigenvalue of the associated $A_{\alpha}$-matrix. In this paper, some lower and upper bounds on $A_{\alpha}$-spread are obtained, which extend the results of $A$-spread and $Q$-spread. Moreover, the trees with the minimum and the maximum $A_{\alpha}$-spread are determined, respectively.

In 2021, the Sombor index was introduced by Gutman, which is a new vertex-degree-based topological index. The Sombor index of a graph G is defined as , where dG(υ) is the degree of the vertex υ in G. Let  and  be the set of trees and unicyclic graphs on n vertices with fixed matching number m, respectively. In this paper, the trees and the unicyclic graphs with the maximum Sombor index among  and  are characterized, respectively.

In this paper, we obtain the quantitative calculation formula of the degree-based topological indices of four standard products for the path and regular graphs, which unify to solve the question on the product of these basic graphs without dealing with it one by one separately. As applications, we give the corresponding calculation formula of the general Randić index, the first general Zagreb index, and the general sum-connectivity index.