Abstract
The -matrix of a graph G is defined as the convex linear combination of the adjacency matrix
and the diagonal matrix of degrees
, i.e.
with
. The maximum modulus among all
-eigenvalues is called the
-spectral radius. In this paper, we order the connected graphs with size m and diameter (at least) d from the second to the
th regarding to the
-spectral radius for
. As by-products, we identify the first
largest trees of order n and diameter (at least) d in terms of their
-spectral radii, and characterize the unique graph with at least one cycle having the largest
-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.
AMS SUBJECT CLASSIFICATION:
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).