Let G be a simple, connected graph and let A(G) the adjacency matrix of G. If D(G) is diagonal vertex degrees G, then for every real $$\alpha \in [0,1]$$ , $$A_{\alpha }(G)$$ defined as }(G) = \alpha + (1- ) A(G).$$ The eigenvalues form }$$ -spectrum $$G_1 {\dot{\vee }} G_2$$ {\underline{\vee \langle \text {v} \rangle {e} denote subdivision-vertex join, subdivision-edge R-vertex join R-edge two...

We study random k-lifts of large, but otherwise arbitrary graphs G. We prove that, with high probability, all eigenvalues of the adjacency matrix of the lift that are not eigenvalues of G are of the order of √ ΔG ln(kn), where ΔG is the maximum degree of G. Similarly, and also with high probability, the “new” eigenvalues of the normalized Laplacian of G are all in an interval of length O( √ ln(...

In this paper, some inequality relations between the Laplacian/signless Laplacian H-eigenvalues and the clique/coclique numbers of uniform hypergraphs are presented. For a connected uniform hypergraph, some tight lower bounds on the largest Laplacian H+-eigenvalue and signless Laplacian H-eigenvalue related to the clique/coclique numbers are given. And some upper and lower bounds on the clique/...

One of the fundamental properties of a graph is the number of distinct eigenvalues of its adjacency or Laplacian matrix. Determining this number is of theoretical interest and also of practical impact. Graphs with small spectra exhibit many symmetry properties and are well suited as interconnection topologies. Especially load balancing can be done on such interconnection topologies in a small n...

A certain signed adjacency matrix of the hypercube, which Hao Huang used last year to resolve sensitivity conjecture, is closely related unique, 4-cycle free, 2-fold cover hypercube. We develop a framework in this connection natural first example relationship between group labeled matrices with few eigenvalues, and combinatorially interesting covering graphs. In particular, we define two-eigenv...

Our first aim in this note is to prove some inequalities relating the eigenvalues of a Hermitian matrix with the eigenvalues of its principal matrices induced by a partition of the index set. One of these inequalities extends an inequality proved by Hoffman in [9]. Secondly, we apply our inequalities to estimate the eigenvalues of the adjacency matrix of a graph, and prove, in particular, that ...

This talk is based primarily on the paper Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees by Marcus, Spielman, and Srivastava [6]. By proving a variant of a conjecture of Bilu and Linial [1], they show that there exist infinite families of d-regular bipartite Ramanujan graphs for d ≥ 3. Of particular interest in the paper by Marcus, Spielman, and Srivastava is their “method of...

We consider a connected threshold graph G with A, S as its adjacency matrix and Seidel respectively. In this paper several spectral properties of are analysed. compute the characteristic polynomial determinant S. A formula for multiplicity eigenvalues $$\pm 1$$ characterisation graphs at most five distinct derived. Finally it is shown that two non isomorphic may be cospectral

We discuss progress on the problem of asymptotically describing the complex homogeneous adjacency eigenvalues of random and complete uniform hypergraphs. There is a natural conjecture arising from analogy to random matrix theory that connects these spectra to that of the all-ones hypermatrix. Several of the ingredients along a possible path to this conjecture are established, and may be of inde...