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.

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 $G$.

Let $t_p(G)$ denote the number of paths in a graph $G$ and let $f:E\rightarrow \mathbb{Z}^+$ be an edge labeling of $G$. The weight of a path $P$ is the sum of the labels assigned to the edges of $P$. If the set of weights of the paths in $G$ is $\{1,2,3,\dots,t_p(G)\}$, then $f$ is called a Leech labeling of $G$ and a graph which admits a Leech labeling is called a Leech graph. In this paper, we prove that the complete bipartite GRAPHS $K_{2,n}$ and $K_{3,n}$ are not Leech GRAPHS and determine the maximum possible value that can be given to an edge in the Leech labeling of a cycle.

A coalition in a graph G = (V, E) consists of two disjoint sets V1 and V2 of vertices, such that neither V1 nor V2 is a dominating set, but the union V1 , V2 is a dominating set of G. A coalition partition in a graph G of order n = |V| is a vertex partition π,= {V1, V2, …, , Vk} such that every set Vi either is a dominating set consisting of a single vertex of degree n-1, or is not a dominating set but forms a coalition with another set Vj. Associated with every coalition partition π,of a graph G is a graph called the coalition graph of G with respect to π, , denoted CG(G,π, ), the vertices of which correspond one-to-one with the sets V1,V2,…, , Vk of π,and two vertices are adjacent in CG(G,π,) if and only if their corresponding sets in π,form a coalition. In this paper, we initiate the study of coalition GRAPHS and we show that every graph is a coalition graph.

Given a graph G, a graph F is said to be Ramsey for G if in every edge coloring of F with two colors, there exists a monochromatic copy of G. The minimum number of edges of a graph F which is Ramsey for G is called the size-Ramsey number of G and is denoted by ^r(G). In 1983, Beck gave a linear upper bound (in terms of n) for ^r(Pn), where Pn is a path on n vertices, giving a positive answer to a question of Erd}os. After that, different approaches were attempted by several authors to reduce the upper bound for ^r(Pn) for sufficiently large n and most of these approaches are based on the classic models of random GRAPHS. Also, Haxell and Kohayakama in 1994 proved that the size Ramsey number of the cycle Cn is linear in terms n, however the Szemeredi's regularity lemma is used in their proof and so no speci c constant coefficient is provided. Here, we provide a method to obtain an upper bound for the size Ramsey number of a graph using good expander GRAPHS such as Ramanujan GRAPHS. In particular, we give an alternative proof for the linearity of the size Ramsey number of paths and cycles. Our method has two privileges in compare to the previous ones. Firstly, it proves the upper bound for every positive integer n in comparison to the random graph methods which needs n to be sufficiently large. Also, due to the recent explicit constructions for bipartite Ramanujan GRAPHS by Marcus, Spielman and Srivastava, we can constructively  nd the GRAPHS with small sizes which are Ramsey for a given graph. We also obtain some results about the bipartite Ramsey numbers.