The general spectral radii of (multicone-)graphs with prescribed degree sequence

Abstract

In the last decade, several scholars proposed an unifying approach to study the spectral theories of the adjacency, Laplacian and signless Laplacian of graphs. The most general graph matrix is the universal adjacency matrix U = αA + βD + γJ + δI, where A, D, J, and I are the adjacency matrix of G, the degree matrix of G, the all-ones matrix, the identity matrix, respectively. Here, we consider $M_{\text{β}}=A+\text{β}D$, with β ≥ 0, and we study the graphs belonging to some given class Γ maximizing the corresponding spectral radius ${\rho }_{\text{β}}$. In particular, we consider connected graphs with prescribed c-cyclic degree sequence, c ∈ {0, 1, 2}, and the multicone graphs defined over them, where the multicone graph is the join of a clique with a given graph. The aim of this paper is to provide the best possible generalization of results to the spectral radius of $M_{\text{β}}$ (and the graph matrices related to it) of several well-known results for multicone graphs over connected graphs with prescribed c-cyclic degree sequence, where c ∈ {0, 1, 2}.

Keywords:

• General spectral radius
• $A_{\alpha }$-spectral radius
• maximal graph
• t-cone graph
• degree sequence
• majorization

Mathematics Subject Classification 2010:

• 05C50
• 15A18

Acknowledgements

The authors would like to thank the referees for their valuable comments which lead to an improvement of the original manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by NNSF of China [grant numbers 11971180, 11501139], the Guangdong Provincial Natural Science Foundation [grant number 2019A1515012052], Characteristic Innovation Project of General Colleges and Universities in Guangdong Province [grant number 2019GKTSCX001], and the Key Project at School Level of Guangzhou Civil Aviation College [No. 18X0429].