The forgotten topological index of a molecular graph G is defined as F (G) = S nÎv (G) d (v)3, where d (v) denotes the degree of vertex v in G. the first through the sixth smallest forgotten indices among all trees, the first through the third smallest forgotten indices among all connected graph with cyclomatic number g = 1, 2, the first through the fourth for g = 3, and the first and the second for g = 4, 5 are determined. These results are compared with those obtained for the first Zagreb index.

We determine the forbidden induced subgraphs for the intersection of the classes of chordal bipartite graphs and line graphs of acyclic directed graphs. This is a first step towards finding the forbidden induced subgraphs for the class of line graphs of directed graphs.

A set W ,V (G) is called a resolving set, if for every two distinct vertices u,v 2 V (G) there exists w 2 W such that d(u, w) 6= d(v, w), where d(x,y) is the distance between the vertices x and y. A resolving set for G with minimum cardinality is called a metric basis. A graph with a unique metric basis is called a uniquely dimensional graph. In this paper, we establish a family of graph called Solis graph, and we prove that if G is a minimal edge unique base graph with the base of size two, then G belongs to the Solis graphs family. Finally, an algorithm is given for , nding the metric dimension of a Solis graph.

An efficient catalyst-free and diastereoselective synthesis of novel dihydropyrano [2,3-d] pyrimidine and dihydropyrano [3,2-c] chromen-annulated benzosultams is described. A number of (E)-N-(2-formylphenyl)-N-methyl-2-phenylethenesulfonamides were synthesized and underwent a one-pot domino Knoevenagel hetero-Diels–Alder reaction, respectively, with N,N-dimethylbarbituric acid and 4-hydroxycoumarin in water, giving the desired products, in moderate to excellent yields.

The Hosoya index $Z(G)$ of a graph $G$ is the total number of matchings in it. In this paper, the recursive formulas of the Hosoya index of semitotal graph $Q(G)$ and total graph $T(G)$ for certain graphs $G$ are obtained. Moreover, we obtain the bounds of the Hosoya index of semitotal and total graphs of a connected graph $G$.

In this paper the p-graphs Matroeids and Hyper graphs and their Matrices are considered. Also it has been shown that a Matroeid and hyper graph are a p-graph and that there is a partial ordered relation between the nodes of the graphs.