The distance eigenvalues of a connected graph G are the eigenvalues of its distance matrix D, and they form the distance spectrum of G. A graph is called distance integral if its distance spectrum consists entirely of integers. We show that no nontrivial tree can be distance integral. We characterize distance integral graphs in the classes of graphs similar to complete split graphs, which, toge...

The Detour matrix (DD) of a graph has for its ( i , j) entry the length of the longest path between vertices i and j. The DD-eigenvalues of a connected graph G are the eigenvalues for its detour matrix, and they form the DD-spectrum of G. The DD-energy EDD of the graph G is the sum of the absolute values of its DDeigenvalues. Two connected graphs are said to be DDequienergetic if they have equa...

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 ...

Abstract. A signed graph Γ = (G, σ) consists of an unsigned graph G = (V, E) and a mapping σ : E → {+,−}. Let Γ be a connected signed graph and L(Γ),L(Γ) be its Laplacian matrix and normalized Laplacian matrix, respectively. Suppose μ1 ≥ · · · ≥ μn−1 ≥ μn ≥ 0 and λ1 ≥ · · · ≥ λn−1 ≥ λn ≥ 0 are the Laplacian eigenvalues and the normalized Laplacian eigenvalues of Γ, respectively. In this paper, ...

In a signed graph G, a negative clique is a complete subgraph having negative edges only. In this article, we give characteristic polynomial expressions, and eigenvalues of some signed graphs having negative cliques. This includes signed cycle graph, signed path graph, a complete graph with disjoint negative cliques, and star block graph with negative cliques.

In this paper, we explore what happens when the same techniques are applied to the problem of estimating eigenvalues of the adjacency operator on finite graphs of bounded degree. In Theorem 7, we show how eigenvalues of the adjacency operator on a finite graph Γ may be bounded in terms of the biggest eigenvalues of the adjacency operator on “geodesic balls” in Γ. We find explicit bounds for the...

Eigenvectors and eigenvalues of discrete graph Laplacians are often used for manifold learning and nonlinear dimensionality reduction. It was previously proved by Belkin and Niyogi [3] that the eigenvectors and eigenvalues of the graph Laplacian converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator of the manifold in the limit of infinitely many data points sampled ind...