We obtain a sharp upper bound for the spectral radius of a nonnegative matrix. This result is used to present upper bounds for the adjacency spectral radius, the Laplacian spectral radius, the signless Laplacian spectral radius, the distance spectral radius, the distance Laplacian spectral radius, the distance signless Laplacian spectral radius of an undirected graph or a digraph. These results are new or generalize some known results., Lihua You, Yujie Shu, Xiao-Dong Zhang., and Obsahuje seznam literatury
We consider the primitive two-colored digraphs whose uncolored digraph has $n+s$ vertices and consists of one $n$-cycle and one $(n-3)$-cycle. We give bounds on the exponents and characterizations of extremal two-colored digraphs.
We obtain upper bounds for generalized indices of matrices in the class of nearly reducible Boolean matrices and in the class of critically reducible Boolean matrices, and prove that these bounds are the best possible.
The symbol K(B,C) denotes a directed graph with the vertex set B∪C for two (not necessarily disjoint) vertex sets B,C in which an arc goes from each vertex of B into each vertex of C. A subdigraph of a digraph D which has this form is called a bisimplex in D. A biclique in D is a bisimplex in D which is not a proper subgraph of any other and in which B ≠ ∅ and C ≠ ∅. The biclique digraph C→ (D) of D is the digraph whose vertex set is the set of all bicliques in D and in which there is an arc from K(B1, C1) into K(B2, C2) if and only if C1 ∩ B2 = ∅. The operator which assigns C→ (D) to D is the biclique operator C→ . The paper solves a problem of E. Prisner concerning the periodicity of C→ .
We examine iteration graphs of the squaring function on the rings $\mathbb{Z}/n\mathbb{Z}$ when $n = 2^{k}p$, for $p$ a Fermat prime. We describe several invariants associated to these graphs and use them to prove that the graphs are not symmetric when $k=3$ and when $k\ge 5$ and are symmetric when $k = 4$.