This paper reports the characteristics of and mutual relationships between various combinatorial structures that give rise to certain imprimitive nonsymmetric three-class association schemes. Nontrivial relation graphs of an imprimitive symmetric 2-class association scheme are m ◦ Kr, (the union of m copies of the complete graph on r vertices) and its complement m ◦Kr, (the complete m-partite s...

Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A (k, l)-kernel N of D is a k-independent set of vertices (if u, v ∈ N then d(u, v), d(v, u) ≥ k) and l-absorbent (if u ∈ V (D) − N then there exists v ∈ N such that d(u, v) ≤ l). A k-kernel is a (k, k − 1)-kernel. A digraph D is cyclically k-partite if there exists a partition {Vi} i=0 of V (D) suc...

A well-known conjecture, often attributed to Ryser, states that the cover number of an r-partite r-uniform hypergraph is at most r?1 times larger than its matching number. Despite considerable effort, particularly in intersecting case, this conjecture remains wide open, motivating pursuit variants original conjecture. Recently, Bustamante and Stein and, independently, Király Tóthmérész consider...

For an ordered subset W={w1,w2,…,wk} of vertices and a vertex v in connected graph G, the k-vector r(v|W)=(d(v,w1),d(v,w2),…,d(v,wk)) is called representation with respect to W, where d(v,wi) distance between wi, for 1≤i≤k. The set W resolving G if r(u|W)≠r(v|W), every pair u,v∈V(G). minimum positive integer k which has cardinality metric dimension denoted as dim(G). A dim(G) basis G. bipartite...

5 We give an optimal degree condition for a tripartite graph to have 6 a spanning subgraph consisting of complete graphs of order 3. This 7 result is used to give an upper bound of 2∆ for the strong chromatic 8 number of n vertex graphs with ∆ ≥ n/6. 9

If G is a graph on n vertices and r 2 2, we let m,(G) denote the minimum number of complete multipartite subgraphs, with r or fewer parts, needed to partition the edge set, f(G). In determining m,(G), we may assume that no two vertices of G have the same neighbor set. For such reduced graphs G, w e prove that m,(G) 2 log,(n + r l)/r. Furthermore, for each k 2 0 and r 2 2, there is a unique redu...

In this paper we study the length of the longest induced cycle in the unit circulant graph Xn = Cay(Zn; Z∗n), where Z∗n is the group of units in Zn. Using residues modulo the primes dividing n, we introduce a representation of the vertices that reduces the problem to a purely combinatorial question of comparing strings of symbols. This representation allows us to prove that the multiplicity of ...

We investigate certain structural properties of random interdependent networks. We start by studying a property known as r-robustness, which is a strong indicator of the ability of a network to tolerate structural perturbations and dynamical attacks. We show that random k-partite graphs exhibit a threshold for r-robustness, and that this threshold is the same as the one for the graph to have mi...

An extremal graph for a graph H on n vertices is a graph on n vertices with maximum number of edges that does not contain H as a subgraph. Let Tn,r be the Turán graph, which is the complete r-partite graph on n vertices with part sizes that differ by at most one. The well-known Turán Theorem states that Tn,r is the only extremal graph for complete graph Kr+1. Erdős et al. (1995) determined the ...

Many distributed graph computing systems have been developed recently for efficient processing of massive graphs. These systems require many messages to be exchanged among computing machines at each step of the computation, making communication bandwidth a major performance bottleneck. We present a coded computing framework that systematically injects redundancy in the computation phase to enab...