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The vertices of a graph are to be labelled with 0 or 1 such that each edge gets the label as the product of its end vertices. If the number of vertices labelled with 0’s and 1’s differ by at most one and if the number of edges labelled with 0’s and 1’s differ by at most by one, then the labelling is called product cordial labelling. Complete characterizations of product cordial dragon graphs is given. We also characterize dragon graphs whose line graphs are product cordial. [ABSTRACT FROM AUTHOR]

Suppose each edge of a simple connected undirected graph is given a unique number from the numbers 1, 2, . . ., q, where q is the number of edges of that graph. Then each vertex is labelled with sum of the labels of the edges incident to it. If no two vertices have the same label, then the graph is called an antimagic graph. We prove that the Cartesian product of wheel graph and path graph is antimagic. [ABSTRACT FROM AUTHOR]

Let G=(E(G),V(G)) be a (molecular) graph with vertex set V(G) and edge set E(G). The forgotten Zagreb index and the hyper Zagreb index of G are defined by F(G)=∑u∈V(G)d(u)3 and HM(G)=∑uv∈E(G)(d(u)+d(v))2 where d(u) and d(v) are the degrees of the vertices u and v in G, respectively. A recent problem called the inverse problem deals with the numerical realizations of topological indices. We see that there exist trees for all even positive integers with F(G)>88 and with HM(G)>158. Along with the result, we show that there exist no trees with F(G)<90 and HM(G)<160 with some exceptional even positive integers and hence characterize the forgotten Zagreb index and the hyper Zagreb index for trees. [ABSTRACT FROM AUTHOR]

Let L(R) denote the set of all non-trivial left ideals of a ring R. The intersection graph of ideals of a ring R is an undirected simple graph denoted by G(R) whose vertices are in a one-to-one correspondence with L(R) and two distinct vertices are joined by an edge if and only if the corresponding left ideals of R have a non-zero intersection. The ideal structure of a ring reflects many ring theoretical properties. There is so much research that has been conducted during the last decade to explore the properties of G(R). This is a survey of the developments in the study on the intersection graphs of ideals of rings since its introduction in 2009. [ABSTRACT FROM AUTHOR]

The concept of super line graph was introduced in the year 1995 by Bagga, Beineke and Varma. Given a graph with at least r edges, the super line graph of index r, Lr(G), has as its vertices the sets of r-edges of G, with two adjacent if there is an edge in one set adjacent to an edge in the other set. The line completion number lc(G) of a graph G is the least positive integer r for which Lr(G) is a complete graph. In this paper, we find the line completion number of grid graph Pn Pm for various cases of n and m. [ABSTRACT FROM AUTHOR]

The concept of Lucky k-polynomials and in particular Lucky X-polynomials was recently introduced. This paper introduces Stirling number of the fourth kind and Lucky partitions of a finite set in order to determine either the Lucky k- or Lucky X-polynomial of a graph. The integer partitions influence Stirling partitions of the second kind. [ABSTRACT FROM AUTHOR]

The concept of super line graph was introduced in the year 1995 by Bagga, Beineke and Varma. Given a graph with at least r edges, the super line graph of index r, Lr(G), has as its vertices the sets of r-edges of G, with two adjacent if there is an edge in one set adjacent to an edge in the other set. The line completion number lc(G) of a graph G is the least positive integer r for which Lr(G) is a complete graph. In this paper, we find the line completion number of grid graph Pn Pm for various cases of n and m. [ABSTRACT FROM AUTHOR]