The Shields-Harary numbers are a class of graph parameters thatmeasure a certain kind of robustness of a graph, thought of as a network of fortified reservoirs, with reference to a given cost function. We prove a result about the Shields-Harary numbers with respect to concave continuous cost functions which will simplify the calculation of these numbers for certain classes of graphs, including ...

Frank Harary contributed numerous questions to a variety of topics in graph theory. One of his favourite topics was the Reconstruction Problem which, in its first issue in 1977, the Journal of Graph Theory described as the major unsolved problem in the field. Together with Plantholt, Frank Harary initiated the study of reconstruction numbers of graphs. We shall here present a survey of some of ...

There are a number of positional games known on the infinite chessboard. One of the most studied is the 5-in-a-row, whose rules are almost identical to the ancient Japanese Go-Moku. Along this line Harary asked if a player can achieve a translated copy of a given polymino P when the two players alternately take the squares of the board. Here we pose his question for general subsets of the board...

The eccentric sequence of a connected graph \(G\) is the nondecreasing eccentricities its vertices. Wiener index sum distances between all unordered pairs vertices \(G\). unique trees that minimise among with given were recently determined by present authors. In this paper we show these results hold not only for index, but large class distance-based topological indices which term Wiener-type in...

The reciprocal distance Laplacian matrix of a connected graph G is defined as RDL(G)=RT(G)−RD(G), where RT(G) the diagonal degrees and RD(G) Harary matrix. Clearly, RDL(G) real symmetric matrix, we denote its eigenvalues λ1(RDL(G))≥λ2(RDL(G))≥…≥λn(RDL(G)). largest eigenvalue λ1(RDL(G)) RDL(G), denoted by λ(G), called spectral radius. In this paper, obtain several upper bounds for sum k in terms...

the stability of a communication network composed of processing nodes and communication links is of prime importance to network designers. as the network begins losing links or nodes, eventually there is a loss in its effectiveness. thus, communication networks must be constructed to be as stable as possible, not only with respect to the initial disruption, but also with respect to the possible...

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.

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...