For a graph G of order n and with eigenvalues λ1 > · · · > λn, the HL-index R(G) is defined as R(G) = max { |λb(n+1)/2c|, |λd(n+1)/2e| } . We show that for every connected bipartite graph G with maximum degree ∆ > 3, R(G) 6 √ ∆− 2 unless G is the the incidence graph of a projective plane of order ∆− 1. We also present an approach through graph covering to construct infinite families of bipartit...

We give several old and some new applications of eigenvalue interlacing to matrices associated to graphs. Bounds are obtained for characteristic numbers of graphs, such as the size of a maximal (colclique, the chromatic number, the diameter, and the bandwidth, in terms of the eigenvalues of the Standard adjacency matrix or the Laplacian matrix. We also deal with inequalities and regularity resu...

Let λ1 be the greatest eigenvalue and λn the least eigenvalue of the adjacency matrix of a connected graph G with n vertices, m edges and diameter D. We prove that if G is nonregular, then Δ− λ1 > nΔ− 2m n(D(nΔ− 2m)+ 1) 1 n(D + 1) , where Δ is the maximum degree of G. The inequality improves previous bounds of Stevanović and of Zhang. It also implies that a lower bound on λn obtained by Alon an...

Let Γ be a distance-regular graph with diameter d and Kneser graph K = Γd, the distance-d graph of Γ. We say that Γ is partially antipodal when K has fewer distinct eigenvalues than Γ. In particular, this is the case of antipodal distance-regular graphs (K with only two distinct eigenvalues), and the so-called half-antipodal distance-regular graphs (K with only one negative eigenvalue). We prov...

Let G be a simple graph with vertex set V (G) = {1, 2, . . . , n} and (0, 1)adjacency matrix A. The eigenvalue μ of A is said to be a main eigenvalue of G if the eigenspace E(μ) is not orthogonal to the all-1 vector j. An eigenvector x is a main eigenvector if xj 6= 0. The main eigenvalues of the connected graphs of order ≤ 5 are listed in [12, Appendix B], and those of all the connected graphs...

By introducing a weight function to the Laplace operator, Bakry and Émery defined the “drift Laplacian” to study diffusion processes. Our first main result is that, given a Bakry–Émery manifold, there is a naturally associated family of graphs whose eigenvalues converge to the eigenvalues of the drift Laplacian as the graphs collapse to the manifold. Applications of this result include a new re...

OF THE DISSERTATION Eigenvalues and Structures of Graphs

We consider weighted graphs, where the edge weights are positive definite matrices. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. We obtain another upper bound which is sharp on the spectral radius of the adjacency matrix and compare with some known upper bounds with the help of some examples of graphs. We also characterize graphs for which the bound is attained.