Let G = (V,E) be a graph of order n and let D ⊆ {0, 1, 2, 3, . . .}. For v ∈ V, let ND(v) = {u ∈ V : d(u, v) ∈ D}. The graph G is said to be D-vertex magic if there exists a bijection f : V (G) → {1, 2, . . . , n} such that for all v ∈ V, ∑ u∈ND(v) f(u) is a constant, called D-vertex magic constant. O’Neal and Slater have proved the uniqueness of the D-vertex magic constant by showing that it c...

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...

In this paper, we define distance partition of vertex set of a graph G with reference to a vertex in it and with the help of the same, a graph with metric dimension two (i.e. 2 ) ( = G β ) is characterized. In the process, we develop a polynomial time algorithm that verifies if the metric dimension of a given graph G is two. The same algorithm explores all metric bases of graph G whenever 2 ) (...

We address the following problem: Given a complete k-partite geometric graph K whose vertex set is a set of n points in R, compute a spanner of K that has a “small” stretch factor and “few” edges. We present two algorithms for this problem. The first algorithm computes a (5 + )-spanner of K with O(n) edges in O(n log n) time. The second algorithm computes a (3 + )-spanner of K with O(n log n) e...

The p-spectral radius of a graph G of order n is defined for any real number p > 1 as λ (G) = max 2 ∑ {i,j}∈E(G) xixj : x1, . . . , xn ∈ R and |x1| + · · ·+ |xn| = 1  . The most remarkable feature of λ(p) is that it seamlessly joins several other graph parameters, e.g., λ(1) is the Lagrangian, λ(2) is the spectral radius and λ(∞)/2 is the number of edges. This paper presents solutions to so...

We present a general framework for studying multi-particle entanglement using graph-theoretic methods. We introduce a technique called bicolored merging for determining combinatorial conditions that must be satisfied for two multiparticle states to be comparable under LOCC. We present several results concerning the possibility and impossibility of various state transformations of multi-partite ...

A k-container C(u,v) of a graph G is a set of k disjoint paths joining u to v. A k-container C(u,v) is a k∗-container if every vertex of G is incident with a path in C(u,v). A bipartite graph G is k∗-laceable if there exists a k∗-container between any two vertices u, v from different partite set of G. A bipartite graph G with connectivity k is super laceable if it is i∗-laceable for all i k. A ...

A digraph obtained by replacing each edge of a complete n-partite graph by an arc or a pair of mutually opposite arcs is called a semicomplete n-partite digraph. We call (D)=max16 i6 n{|Vi|} the independence number of the semicomplete n-partite digraph D, where V1; V2; : : : ; Vn are the partite sets of D. Let p and c, respectively, denote the number of vertices in a longest directed path and t...

For an integer t and a fixed graph H, we consider the problem of finding a maximum tmatching not containing H as a subgraph, which we call the H-free t-matching problem. This problem is a generalization of the problem of finding a maximum 2-matching with no short cycles, which has been well-studied as a natural relaxation of the Hamiltonian circuit problem. When H is a complete graph Kt+1 or a ...