The independence ratio i(G) of a graph G is the ratio of its independence number and the number of vertices. The ultimate categorical independence ratio of a graph G is defined as limk→∞ i(G×k), where G×k denotes the kth categorical power of G. This parameter was introduced by Brown, Nowakowski and Rall, who asked about its value for complete multipartite graphs. In this paper we determine the ...

A digraph obtained by replacing each edge of a complete multipartite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete multipartite digraph. L. Volkmann conjectured that l ≤ 2c− 1, where l (c, respectively) is the number of vertices in a longest path (longest cycle) of a strong semicomplete multipartite digraph. The bound on l is sharp. We s...

In this paper, we study the achromatic indices of the regular complete multipartite graphs and obtain the following results: (1) A good upper bound for the achromatic index of the regular complete multipartite graph which gives the exact values of an infinite family of graphs and solves a problem posed by Bouchet. (2) An improved Bouchet coloring which gives the achromatic indices of another in...

We introduce a new graph operator, called the weak factor graph, which is close to the well-known clique-graph operator but which rather operates in terms of bicliques in a multipartite graph, and we address the problem of the convergence of the series of graphs obtained by the iterative application of this operator. As for the clique-graph operator, it is easy to find graphs for which the seri...

We define a new graph operator, called the weak-factor graph, which comes from the context of complex network modelling. The weak-factor operator is close to the well-known clique-graph operator but it rather operates in terms of bicliques in a multipartite graph. We address the problem of the termination of the series of graphs obtained by iteratively applying the weak-factor operator starting...

In 1975 Bollob\'as, Erd\H os, and Szemer\'edi asked the following question: given positive integers $n, t, r$ with $2\le t\le r-1$, what is largest minimum degree $\delta(G)$ among all $r$-partite graphs $G$ parts of size $n$ which do not contain a copy $K_{t+1}$? The $r=t+1$ case has attracted lot attention was fully resolved by Haxell Szab\'{o}, Szab\'{o} Tardos in 2006. this paper we investi...

This thesis is an attempt to enhance understanding of the following questions AGiven a multipartite quantum state (possibly mixed), how to find out whether it is entangled or separable? (Detection of entanglement.) BGiven an entangled state, how to decide how much entangled it is? (Measure of entanglement.), in the context of multipartite quantum states. We have explored two approaches. In the ...

A set S of edge-disjoint hamilton cycles in a graph T is said to be maximal if the hamilton cycles in S form a subgraph of T such that T −E(S) has no hamilton cycle. The set of integers m for which a graph T contains a maximal set of m edge-disjoint hamilton cycles has previously been determined whenever T is a complete graph, a complete bipartite graph, and in many cases when T is a complete m...

The Ramsey number r(F, G) is the least number N such that in every two-coloring (R, B)=(red, blue) of the edges of KN , either there is a red copy of F or else a blue copy of G . The inequality (1) holds in view of the fact that the edges of the complete graph of order (m-1) (n-1) +s-1 can be given the coloring in which the blue graph, denoted (B), is isomorphic to (m-1)K„_~UKS_~ . Then the red...

A digraph obtained by replacing each edge of a complete p-partite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicom-plete p-partite digraph, or just a semicomplete multipartite digraph. A semicomplete multipartite digraph with no cycle of length two is a multipartite tournament. In a digraph D, an r-king is a vertex q such that every vertex in D ...