If G and H are graphs, define the Ramsey number to be the least number p such that if the lines of the complete graph Kp are colored red and blue (say),either the red subgraph contains a copy of G or the blue subgraph contains H. Also set r(G) = r(G,G) ; these are called the diagonal Ramsey numbers. These definitions are taken from Chvátal and Harary [1J ; other terminology will follow Harary [...

A graph G is a combined k-fault-tolerant Hamiltonian graph (also called a combined k-Hamiltonian graph) if G − F is Hamiltonian for every subset F ⊂ (V(G) ∪ E(G)) with |F| = k. A combined k-Hamiltonian graph G with |V(G)| = n is optimal if it has the minimum number of edges among all n-node k-Hamiltonian graphs. Using the concept of node expansion, we present a powerful construction scheme to c...

A graph G is well-covered if all its maximal stable sets have the same size, denoted by α(G) (M. D. Plummer, 1970). If sk denotes the number of stable sets of cardinality k in graph G, and α(G) is the size of a maximum stable set, then I(G;x) = α(G) ∑ k=0 skx k is the independence polynomial of G (I. Gutman and F. Harary, 1983). J. I. Brown, K. Dilcher and R. J. Nowakowski (2000) conjectured th...

In European J. Combin. 30 (2009), 986–995, S.M. Hegde recently introduced set colorings of a graph G as an assignment (function) of distinct subsets of a finite set X of colors to the vertices of G, where the colors of the edges are obtained as the symmetric difference of the sets assigned to their end vertices (which are also distinct). A set coloring is called a proper set coloring if all the...

An independent set in a graph is a set of pairwise non-adjacent vertices, and α(G) is the size of a maximum independent set in the graph G. If sk is the number of independent sets of cardinality k in G, then I(G;x) = s0 + s1x+ s2x 2 + ...+ sαx , α = α (G) , is called the independence polynomial of G (I. Gutman and F. Harary, 1983). If sα−i = f (i) · sα−j holds for every i ∈ {0, 1, ..., bα/2c}, ...

Let $G$ be a connected graph, and let $D[G]$ denote the double graph of $G$. In this paper, we first derive closed-form formulas for different distance based topological indices for $D[G]$ in terms of that of $G$. Finally, as illustration examples, for several special kind of graphs, such as, the complete graph, the path, the cycle, etc., the explicit formulas for some distance based topologica...

abstract. topological indices are the numerical value associated with chemical constitution purporting for correlation of chemical structure with various physical properties, chemical reactivity or biological activity. graph theory is a delightful playground for the exploration of proof techniques in discrete mathematics and its results have applications in many areas of sciences. a graph is a ...

The fact that the properties of a molecule are tightly connected to its structural  characteristics  is one of the fundamental concepts in chemistry. In this connection,  graph theory has been successfully applied in developing some relationships between topological indices and some thermodynamic properties. So ,  a novel method for computing the new descriptors to construct a quantitative rela...