Spectral radius of graphs with given matching number

Abstract

In this paper, we show that of all graphs of order n with matching number β, the graphs with maximal spectral radius are K n if n = 2β or 2β + 1; K 2β+1 ∪ K n−2β−1 if 2β + 2 n < 3β + 2; K β K n−β or K 2β+1 ∪ K n−2β−1 if n = 3β + 2; K β K n−β if n > 3β + 2, where K t is the empty graph on t vertices.

On the methods to assess significance in nestedness analyses

“…Moreover, adding edges to a graph (which corresponds to add presences to a binary matrix) necessarily leads to an increment in its spectral radius (Feng et al 2007). Consequently, the spectral radius scales (independently of nestedness) with matrix size and number of occurrences, for which q(A) of a given matrix should be compared only with the q(A)s of matrices with the same number of occurrences, which is not ensured by the CE null model.…”

The least eigenvalue of a graph with a given domination number

“…Brualdi and Solheid [5] proposed the following problem concerning the spectral radii: Given a set of graphs G, find an upper bound for the spectral radii of graphs in G and characterize the graphs in which the maximal spectral radius is attained. This problem has been well studied, see [3,10,13,20] for example. Recently, researchers have begun to pay attention to the least eigenvalues of graphs with a given value of some well-known integer graph invariant: for instance: order and size [1,2,8,18], unicyclic graphs with a given number of pendant vertices [14], matching number and independence number [21], number of cut vertices [22], connectivity [23], chromatic number [9].…”

Zagreb, Harary and hyper-Wiener indices of graphs with a given matching number

“…In [19] the authors examined the extremal graphs with maximal spectral radius among graphs of order n with matching number β. Zhou and Trinajstić [20] determined the minimum Kirchhoff index of connected graphs in terms of the number of vertices and matching number.…”