Spectral radius and degree sequence of a graph

Abstract

Let G be a simple connected graph of order n with degree sequence d 1 , d 2 , · · · , d n in non-increasing order. The spectral radius ρ(G) of G is the largest eigenvalue of its adjacency matrix. For each positive integer ℓ at most n, we give a sharp upper bound for ρ(G) by a function of d 1 , d 2 , · · · , d ℓ , which generalizes a series of previous results.

Spectral radius of bipartite graphs

“…Remark 3.7. Other previous results shown by the style of the above proof can be found in [17,14,9,13]. Similar earlier results are referred to [6,7,18,11,12].…”

A (Forgotten) Upper Bound for the Spectral Radius of A Graph

“…In 2012, Liu and Weng [12] proved (3) using a different approach. They also proved there is equality if and only if G is regular or there exists 2 ≤ t ≤ k such that d 1 = d t−1 = n − 1 and d t = d n .…”

“…Lemma 2. If G L denotes the line graph of G then: The following lemma is proved in varying ways in [15,5,12].…”

Spectral radius and signless Laplacian spectral radius of strongly connected digraphs

“…The spectral radius and the signless Laplacian spectral radius of undirected graph are well treated in the literature, see [6,7,8,9,10,17,19] and so on, but there is not much known about digraphs. Recently, R.A. Brualdi wrote a stimulating survey on the spectra of digraphs [4].…”

“…Then we give a new sharp upper bound of the signless Laplacian spectral radius among all simple digraphs and compare them with the upper bounds given in [5] as inequalities (1.1)-(1.4). The technique used in the result is motivated by [8,17] et al By the definition of Q(D), the i-th row sum of Q(D) is 2d + i . Follows from Lemma 2.1, we can get q( − → C n ) = 2 and q( From Lemma 2.1 and Corollary 2.5, we easily get the following results.…”