We consider the enumeration of unlabeled directed hypergraphs by using Pólya’s counting theory and Burnside’s counting lemma. Instead of characterizing the cycle index of the permutation group acting on the hyperarc set A, we treat each cycle in the disjoint cycle decomposition of a permutation ρ acting on A as an equivalence class (or orbit) of A under the operation of the group generated by ρ...

We show that the geodetic game, introduced by Fraenkel and Harary, is decidable in polynomial time on AT-free graphs.

Given a directed graph G = (V,E) and an integer k ≥ 1, a k-transitive-closure-spanner (k-TCspanner) of G is a directed graph H = (V,EH) that has (1) the same transitive-closure as G and (2) diameter at most k. In some applications, the shortcut paths added to the graph in order to obtain small diameter can use Steiner vertices, that is, vertices not in the original graph G. The resulting spanne...

Abstract The Estrada index of a graph/network is defined as the trace adjacency matrix exponential. It has been extended to other graph-theoretic matrices, such Laplacian, distance, Seidel adjacency, Harary, etc. Here, we describe many these extensions, including new ones, Gaussian, Mittag–Leffler and Onsager ones. More importantly, contextualize all indices in physico-mathematical frameworks w...

Players A and B alternatively colour edges of a graph G, red and blue respectively. Let Lsym(G) be the largest number of moves during which B can keep the red and blue subgraphs isomorphic, no matter how A plays. This function was introduced by Harary, Slany and Verbitsky who in particular showed that for complete bipartite graphs we have Lsym(Km,n) = mn 2 if mn is even and that Lsym(K2m+1,2n+1...

We prove the Kauffman–Harary Conjecture, posed in 1999: given a reduced, alternating diagram D of a knot with prime determinant p, every non-trivial Fox p-coloring of D will assign different colors to different arcs.

Let G = (V, E) be a graph with v vertices and e edges. An (a, d)-vertex-antimagic total labeling is a bijection λ from V (G) ∪ E(G) to the set of consecutive integers 1, 2, . . . , v + e, such that the weights of the vertices form an arithmetic progression with the initial term a and common difference d. If λ(V (G)) = {1, 2, . . . , v} then we call the labeling a super (a, d)-vertex-antimagic t...

In this paper we extend the context of Steiner ratio and examine the influence of Steiner points on the weight of a graph in two generalizations of the Euclidean minimum weight connected graph (MST). The studied generalizations are with respect to the weight function and the connectivity condition. First, we consider the Steiner ratio of Euclidean minimum weight connected graph under the budget...

Alexandre A. Steiner, Sumana Chakravarty, Jared R. Robbins, Alexander S. Dragic, Jennifer Pan, Miles Herkenham, and Andrej A. Romanovsky Systemic Inflammation Laboratory, Trauma Research, St. Joseph’s Hospital and Medical Center, Phoenix, Arizona; and Section on Functional Neuroanatomy, National Institute of Mental Health, National Institutes of Health, United States Department of Health and Hu...