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.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
