Abstract
Abstract Recently, the extremal problem of the spectral radius in the class of complements of trees, unicyclic graphs, bicyclic graphs and tricyclic graphs had been studied widely. In this paper, we extend the largest ordering of A α -spectral radius among all complements of bicyclic and tricyclic graphs with n vertices, respectively.
Highlights
Throughout this paper, we only concern with simple undirected graphs
We extend the largest ordering of Aα-spectral radius among all complements of bicyclic and tricyclic graphs with n vertices, respectively
The adjacency matrix and diagonal matrix of graph G is denoted by A(G) and D(G), respectively
Summary
Throughout this paper, we only concern with simple undirected graphs. Let G = (V , E) be a graph with vertex set V(G) = {v , v , ..., vn}, and let dG(vi) be the degree of vertex vi of G. We use G and ∆(G) to denote the complement and maximum degree of graph G, respectively. The unique ρα-maximal graph among the class of complements of trees, unicyclic graphs and bicyclic graphs was determined by Zhang and Guo [9], that is, Theorem 1.2. As shown in the later, di erent from the ordering of largest spectral radius of trees, unicyclic graphs, bicyclic graphs and tricyclic graphs [5], we nd the solution to this problem is rather di cult. Α + ) = φ (α) > , as the smallest root of φ (α) is about Combining this with Υ (x) = ( − α)Ψ (x), we have Υ (x) > , and Claim 1 holds.
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