Given a graph G, write μ (G) for the largest eigenvalue of its adjacency matrix, ω (G) for its clique number, and wk (G) for the number of its k-walks. We prove that the inequalities wq+r (G) wq (G) ≤ μ (G) ≤ ω (G) − 1 ω (G) wr (G) hold for all r > 0 and odd q > 0. We also generalize a number of other bounds on μ (G) and characterize pseudo-regular and pseudo-semiregular graphs in spectral terms.

Let G be a graph, χ be its chromatic number, λ be the largest eigenvalue of its Laplacian, and µ be the largest eigenvalue of its adjacency matrix. Then, complementing a well-known result of Hoffman, we show that λ ≥ χ χ − 1 µ with equality holding for regular complete χ-partite graphs. We denote the eigenvalues of a Hermitian matrix A as µ (A) = µ 1 (A) ≥ · · · ≥ µ min (A). Given a graph G, we...

We characterize the graphs which achieve the maximum value of the spectral radius of the adjacency matrix in the sets of all graphs with a given domination number and graphs with no isolated vertices and a given domination number. AMS Classification: 05C35, 05C50, 05C69

We give lower bounds for the spectral radius of nonnegative matrices and nonnegative symmetric matrices, and prove necessary and sufficient conditions to achieve these bounds.

We prove theorems of Perron–Frobenius type for positive elements in partially ordered topological algebras satisfying certain hypotheses. We show how some of our results relate to known results on Banach algebras. We give examples and state some open questions. © 2005 Elsevier Inc. All rights reserved. AMS classification: 47A10; 46H35; 47B65; 15A48

In this note, we present two lower bounds for the spectral radius of the Laplacian matrices of triangle-free graphs. One is in terms of the numbers of edges and vertices of graphs, and the other is in terms of degrees and average 2-degrees of vertices. We also obtain some other related results.

Acknowledgements I first would like to thank my promotor Vincent Blondel for accepting me as his first Ph.D student, and providing me with a challenging research subject. His constructive comments, his pragmatism and his initiative were essential in the realization of this thesis. Several researchers contributed to this thesis. I would like to especially thank Alexander Vladimirov and Yurii Nes...

The independence number of a graph is defined as the maximum size of a set of pairwise non-adjacent vertices and the spectral radius is defined as the maximum eigenvalue of the adjacency matrix of the graph. Xu et al. in [The minimum spectral radius of graphs with a given independence number, Linear Algebra and its Applications 431 (2009) 937–945] determined the connected graphs of order n with...