Ordering of graphs with fixed size and diameter by A_α-spectral radii

Abstract

The $A_{\alpha }$-matrix of a graph G is defined as the convex linear combination of the adjacency matrix $A(G)$ and the diagonal matrix of degrees $D(G)$, i.e. $A_{\alpha }(G)=\mathrm{\alpha D}(G)+(1-\alpha )A(G)$ with $\alpha \in [0,1]$. The maximum modulus among all $A_{\alpha }$-eigenvalues is called the $A_{\alpha }$-spectral radius. In this paper, we order the connected graphs with size m and diameter (at least) d from the second to the $(\lfloor \frac{d}{2}\rfloor +1)$th regarding to the $A_{\alpha }$-spectral radius for $\alpha \in [\frac{1}{2},1)$. As by-products, we identify the first $\lfloor \frac{d}{2}\rfloor$ largest trees of order n and diameter (at least) d in terms of their $A_{\alpha }$-spectral radii, and characterize the unique graph with at least one cycle having the largest $A_{\alpha }$-spectral radius among graphs of size m and diameter (at least) d. Consequently, the corresponding results for signless Laplacian matrix can be deduced as well.

KEYWORDS:

• $A_{\alpha }$-spectral radius
• size
• diameter
• double leading eigenvectors

AMS SUBJECT CLASSIFICATION:

• 05C50

Acknowledgments

The authors would like to express their sincere gratitude to the referee for his or her careful reading and insightful suggestions, which led to a number of improvements to this paper.

Disclosure statement

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

Additional information

Funding

Financially supported by the National Natural Science Foundation of China (Grant No. 12171190).