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In this paper, we study multi-level distance edge labeling for infinite rectangular, hexagonal and triangular grids. We label the edges with non-negative integers. If the edges are adjacent, then their color difference is at least 2 and if they are separated by exactly a single edge, then their colors must be distinct. We find the edge coloring number of these grids to be 9, 7 and 16, respectively so that we could color the edges of a rectangular, hexagonal and triangular grid with at most 10, 8 and 17 colors, respectively using this coloring technique. Repeating the sequence pattern for different grids, we can color the edges of a grid of larger size.

A path decomposition of a graph is a collection of its edge disjoint paths whose union is G. The pendant number Πp is the minimum number of end vertices of paths in a path decomposition of G. In this paper, we determine the pendant number of corona products and rooted products of paths and cycles and obtain some bounds for the pendant number for some specific derived graphs. Further, for any natural number n, the existence of a connected graph with pendant number n has also been established.

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]

Signed graphs are graphs with every edge is signed either positive or negative. Given an n vertex graph, the vertices are bijectively labelled from 1 to n. A signed graph is a sum signed graph if and only if every edge is signed negative whenever the sum of the vertex labels exceeds n and every edge is signed positive whenever the sum of the vertex labels does not exceed n. For a parity signed graph, an edge receives a negative sign, if the end vertices are of opposite parity and a positive sign otherwise. Cordial signed graphs are the ones with the difference between the total number of negative edges and the positive ones is at most 1. We discuss the connection between sum signed labeling with parity signed labeling and cordial labeling. The absolute cordial condition for graphs satisfying sum signed labeling will be analyzed. [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]

In this paper, we study multi-level distance edge labeling for infinite rectangular, hexagonal and triangular grids. We label the edges with non-negative integers. If the edges are adjacent, then their color difference is at least 2 and if they are separated by exactly a single edge, then their colors must be distinct. We find the edge coloring number of these grids to be 9, 7 and 16, respectively so that we could color the edges of a rectangular, hexagonal and triangular grid with at most 10, 8 and 17 colors, respectively using this coloring technique. Repeating the sequence pattern for different grids, we can color the edges of a grid of larger size. [ABSTRACT FROM AUTHOR]

Butane derivatives are chemical compounds formally derived from Butane C4H10 by replacement of one or more hydrogen atoms with other atoms or functional groups. In this paper, we do the QSPR analysis of few Butane derivatives with respect to some selected degree based topological indices and one eccentricity based topological index. In QSPR studies, topological indices are extensively used in determining specific bioactivity of chemical compounds. Our study showcases some important results on the correlation between Heavy atomic count, Complexity, Density, Surface tension and Index of refraction of Butane derivatives with the selected topological indices which further helps in characterizing the predicting power of these topological indices. [ABSTRACT FROM AUTHOR]