The Ramsey number r(H) of a graph H is the minimum integer n such that any two-coloring edges complete Kn contains monochromatic copy H. While this definition only asks for single H, it often case every two-edge-coloring on vertices many copies over all two-colorings Kr(H) will be referred to as threshold multiplicity Addressing problem Harary and Prins, who were first systematically study quan...

The Ramsey number r(H, G) is defined as the minimum N such that for any coloring of the edges of the N-vertex complete graph KN in red and blue, it must contain either a ted H or a blue G. In this paper we show that for any graph G without isolated vertices, r(K,, G)< 2qf 1 where G has q edges. In other words, any graph on 2q+ 1 vertices with independence number at most 2 contains every (isolat...

We study state transfer in quantum walks on graphs relative to the adjacency matrix. Our motivation is to understand how the addition of pendant subgraphs affect state transfer. For two graphs G and H, the Frucht-Harary corona product G ◦H is obtained by taking |G| copies of the cone K1 + H and by identifying the conical vertices according to G. Our work explores conditions under which the coro...

Harary, F. , 1997 Graph Theory, Narosa Publishing House. Arumugam, S. , Joseph, J. P. 1999 On graphs with equal domination and connected domination numbers, Discrete Mathematics vol. 206, 45-49. Deo, N. 2005 Graph Theory with Applications to Engineering and Computer Science, Prentice-Hall of India Private Limited. Saoud, M. , Jebran, J. 2009 Finding A Minimum Dominating Set by Transforming Domi...

The acyclic point-connectivity of a graph G, denoted (G), is the minimum number of points whose removal from G results in an acyclic graph. In a 1975 paper, Harary stated erroneously that (Qn 2 n-I i where Qn denotes the n-cube. We prove that for n > 4, 7 2 n-4 -< () -< 2 n-I 2n-y-2, where y [log2(ni)]. We show that the upper bound is obtained for n -< 8 and conjecture that it is obtained for a...

A graph G has a representation modulo r if there exists an injective map f : V (G) → {0, 1, ..., r − 1} such that vertices u and v are adjacent if and only if f(u) − f(v) is relatively prime to r. The representation number of G, rep(G), is the smallest integer such that G has a representation modulo r. In this paper we study the representation numbers of various graphs, such as the complete ter...

We consider vertex colorings of graphs in which each color has an associated cost which is incurred each time the color is assigned to a vertex. The cost of the coloring is the sum of the costs incurred at each vertex. The cost chromatic number of a graph with respect to a cost set is the minimum number of colors necessary to produce a minimum cost coloring of the graph. We show that the cost c...

We consider bipartite graphs on m red points and n blue points, where m ^ n, and prove that, for any fixed k, almost all such graphs (labelled or unlabelled) are k-connected as n -» oo, provided m > C logn, where C depends on k. If Tmn is the number of such unlabelled graphs, we show that Tmn ~ 2 /(m \n!). If T'mn is the number of such unlabelled graphs with the colours removed, then T'mn ~ Tmn...