Let G be a simple connected graph with vertex set V = {1, 2, . . . , n}. Let d(i, j) denote the distance between vertices i and j. For i ∈ V , the degree of i and the average of the degree of the vertices adjacent to i are denoted by di and mi, respectively. The 2-degree of vertex i is denoted by ti, which is the sum of degrees of the vertices adjacent to i, that is ti = midi. Let Ni be the sum...

We prove the `p-spectral radius formula for n-tuples of commuting Banach algebra elements. This generalizes results of [6], [7] and [10]. Let A be a Banach algebra with the unit element denoted by 1. Let a = (a1, . . . , an) be an n-tuple of elements of A. Denote by σ(a) the Harte spectrum of a, i.e. λ = (λ1, . . . , λn) / ∈ σ(a) if and only if there exist u1, . . . , un, v1, . . . , vn ∈ A suc...

In this paper, we study some extremal problems of three kinds of spectral radii of k-uniform hypergraphs (the adjacency spectral radius, the signless Laplacian spectral radius and the incidence Q-spectral radius). We call a connected and acyclic k-uniform hypergraph a supertree. We introduce the operation of “moving edges” for hypergraphs, together with the two special cases of this operation: ...

The structure of graphs whose largest eigenvalue is bounded by 3 2 √ 2 (≈ 2.1312) is investigated. In particular, such a graph can have at most one circuit, and has a natural quipu structure.

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...

A sign pattern is an n × n matrix, A, with entries in {+,−, 0}. If A is a real n× n matrix for which each entry has the same sign as its corresponding entry in A, then A is a realization of A, and we write A ∈ A. The 2n-conjecture is related to the study of the spectral properties among the matrices in A. The n × n sign pattern A is a spectrally arbitrary pattern (or a SAP, for short), provided...