We prove three results about the spectral radius μ (G) of a graph G : (a) Let Tr (n) be the r-partite Turán graph of order n. If G is a Kr+1-free graph of order n, then μ (G) < μ (Tr (n)) unless G = Tr (n) . (b) For most irregular graphs G of order n and size m, μ (G)− 2m/n > 1/ (2m+ 2n) . (c) Let 0 ≤ k ≤ l. If G is a graph of order n with no K2 +Kk+1 and no K2,l+1, then μ (G) ≤ min {

‎Let $Gamma$ be a graph with adjacency eigenvalues $lambda_1leqlambda_2leqldotsleqlambda_n$‎. ‎Then the energy of‎ ‎$Gamma$‎, ‎a concept defined in 1978 by Gutman‎, ‎is defined as $mathcal{E}(G)=sum_{i=1}^n|lambda_i|$‎. ‎Also‎ ‎the Estrada index of $Gamma$‎, ‎which is defined in 2000 by Ernesto Estrada‎, ‎is defined as $EE(Gamma)=sum_{i=1}^ne^{lambda_i}$‎. ‎In this paper‎, ‎we compute the eigen...

Several inequalities on vertex degrees, eigenvalues, Laplacian eigen-values, and signless Laplacian eigenvalues of graphs are presented in this note. Some of them are generalizations of the inequalities in [2]. We consider only finite undirected graphs without loops or multiple edges. Notation and terminology not defined here follow that in [1]. We use [n] to denote the set of { 1, 2, ..., n}. ...

A homogeneous graph is a graph together with a group which acts transitively on vertices as symmetries of the graph. We consider Laplacians of homogeneous graphs and generalizations of Laplacians whose eigenvalues can be associated with various equilibria of forces in molecules (such as vibrational modes of buckyballs). Methods are given for calculating such eigenvalues by combining concepts an...

A graph Γ with diameter d is strongly distance-regular if Γ is distanceregular and its distance-d graph Γd is strongly regular. The known examples are all the connected strongly regular graphs (i.e. d = 2), all the antipodal distanceregular graphs, and some distance-regular graphs with diameter d = 3. The main result in this paper is a characterization of these graphs (among regular graphs with...

Let Γ be a regular (connected) graph with n vertices and d + 1 distinct eigenvalues. As a main result, it is shown that Γ is an r-antipodal distanceregular graph if and only if the distance graph Γd is constituted by disjoint copies of the complete graph Kr, with r satisfying an expression in terms of n and the distinct eigenvalues. AMS subject classifications. 05C50 05E30

In this paper, we derive an upper bound for higher eigenvalues of the normalized Laplace operator associated with a symmetric finite graph in terms lower eigenvalues.