Write (A) = 1 (A) min (A) for the eigenvalues of a Hermitian matrix A. Our main result is: let A be a Hermitian matrix partitioned into r r blocks so that all diagonal blocks are zero. Then for every real diagonal matrix B of the same size as A; (B A) B + 1 r 1 : Let G be a nonempty graph, (G) be its chromatic number, A be its adjacency matrix, and L be its Laplacian. The above inequality impli...

Abstract In this paper, we give the spectrum of a matrix by using the quotient matrix, then we apply this result to various matrices associated to a graph and a digraph, including adjacency matrix, (signless) Laplacian matrix, distance matrix, distance (signless) Laplacian matrix, to obtain some known and new results. Moreover, we propose some problems for further research. AMS Classification: ...

Given a graph G = (V,E) and a vertex u ∈ V , we would like to explore every vertex (and every edge) reachable from u. Let n = |V | and m = |E|. The graph may be directed or undirected. We will assume that the graph is given in its adjacency list representation (rather than adjacency matrix), i.e. for each vertex u, we have a linked list adj[u] of the neighbors of u (all vertices v such that (u,...

A multiplicative attribute graph is a random graph in which vertices are represented by random words of length t in a finite alphabet Γ, and the probability of adjacency is a symmetric function Γt×Γt → [0, 1]. These graphs are a generalization of stochastic Kronecker graphs, and both classes have been shown to exhibit several useful real world properties. We establish asymptotic bounds on the s...

Notation: λ1(A) = largest eigenvalue of A. Motivation: The largest eigenvalue tells us the spectrum of a matrix and thus can be somewhat useful. More crucially for the adjacency matrix of a graph, we know exactly that the all one’s vector is the largest eigenvector, and thus by working orthogonal to this vector we can approximate the second largest eigenvalue of the graph, which tells us how we...

A local property reconstructor for a graph property is an algorithm which, given oracle access to the adjacency list of a graph that is “close” to having the property, provides oracle access to the adjacency matrix of a “correction” of the graph, i.e. a graph which has the property and is close to the given graph. For this model, we achieve local property reconstructors for the properties of co...

One of the most relevant tasks in network analysis is the detection of community structures, or clustering. Most popular techniques for community detection are based on the maximization of a quality function called modularity, which in turn is based upon particular quadratic forms associated to a real symmetric modularity matrix M , defined in terms of the adjacency matrix and a rank one null m...

We study space and time efficient quantum algorithms for two graph problems – deciding whether an n-vertex graph is a forest, and whether it is bipartite. Via a reduction to the s-t connectivity problem, we describe quantum algorithms for deciding both properties in Õ(n) time and using O(log n) classical and quantum bits of storage in the adjacency matrix model. We then present quantum algorith...

Let G = (V,E) be a graph with vertex set V = {1, . . . , n} and edge set E. Throughout the paper, a graph G is undirected and simple (i.e., has no multi-edges or loops). We allow G to be disconnected. The Laplacian of G is the matrix L(G) = D−A, where D is the diagonal matrix whose entries are the degree of the vertices and A is the adjacency matrix of G. Chung’s normalized Laplacian L̃(G) [6] i...