A graph is called integral if all eigenvalues of its adjacency matrix are integers. In this paper, we investigate integral complete r-partite graphsKp1,p2,...,pr =Ka1·p1,a2·p2,...,as ·ps with s=3, 4.We can construct infinite many new classes of such integral graphs by solving some certain Diophantine equations. These results are different from those in the existing literature. For s = 4, we giv...

Abstract. Two central topics of study in combinatorics are the so-called evolution of random graphs, introduced by the seminal work of Erdős and Rényi, and the family of H-free graphs, that is, graphs which do not contain a subgraph isomorphic to a given (usually small) graph H . A widely studied problem that lies at the interface of these two areas is that of determining how the structure of a...

The chromatic number of a graph G, denoted by χ(G), is the minimum number of colors such that G can be colored with these colors in such a way that no two adjacent vertices have the same color. A clique in a graph is a set of mutually adjacent vertices. The maximum size of a clique in a graph G is called the clique number of G. The Turán graph Tn(k) is a complete k-partite graph whose partition...

For a fixed r, let fr(n) denote the minimum number of complete r-partite rgraphs needed to partition the complete r-graph on n vertices. The Graham-Pollak theorem asserts that f2(n) = n − 1. An easy construction shows that fr(n) 6 (1 + o(1)) ( n br/2c ) , and we write cr for the least number such that fr(n) 6 cr(1 + o(1)) ( n br/2c ) . It was known that cr < 1 for each even r > 4, but this was ...

The Erdős–Hajnal Theorem asserts that non-universal graphs, is, graphs do not contain an induced copy of some fixed graph H, have homogeneous sets size significantly larger than one can generally expect to find in a graph. We obtain two results this flavor the setting r-uniform hypergraphs. A theorem Rödl if n-vertex is then it contains almost set (i.e. with edge density either very close 0 or ...

An n-tournament is an orientation of a complete n-partite graph. It was proved by J.A. Bondy in 1976 that every strongly connected n-partite tournament has an n-cycle. We characterize strongly connected n-partite tournaments in which a longest cycle is of length n and, thus, settle a problem in L. Volkmann, Discrete Math. 245 (2002) 19-53.

We give sufficient conditions for a tripartite graph to have a spanning subgraph consisting of complete graphs of order 3. This is applied to bound the strong chromatic number of a graph.

In this paper, the exact formulae for the generalized degree distance, degree distance and reciprocal degree distance of strong product of a connected and the complete multipartite graph with partite sets of sizes m0, m1, . . . , mr&minus1 are obtained. Using the results obtained here, the formulae for the degree distance and reciprocal degree distance of the closed and open fence graphs are co...