A Variation of Decomposition Under a Length Constraint

Introducing and initiating a study of a new variation of decomposition namely equiparity induced path decomposition of a graph which is defined to be a decomposition in which all the members are induced paths having same parity

On Color Energy of Few Classes of Bipartite Graphs and Corresponding Color Complements

For a given colored graph G, the color energy is defined as Ec(G) = Σλi, for i = 1, 2,…., n; where λi is a color eigenvalue of the color matrix of G, Ac (G) with entries as 1, if both the corresponding vertices are neighbors and have different colors; -1, if both the corresponding vertices are not neighbors and have same colors and 0, otherwise. In this article, we study color energy of graphs with proper coloring and L (h, k)-coloring. Further, we examine the relation between Ec(G) with the corresponding color complement of a given graph G and other graph parameters such as chromatic number and domination number. AMS Subject Classification: 05C15, 05C5

On Equality and Strong Equality of Domination Number and Independent Domination Number in Graphs

 In this paper we explore graphs having same domination number and independent domination number . Such graphs are denoted as ( , )-graphs. Several families of  ( , )-graphs have been constructed. The realization problem for graphs with  =  = a for any given positive integer a has been solved. Furthermore, properties of graphs in which every  -set is a -set has been investigated

Induced label graphoidal graphs

Abstract. Let G be a non-trivial, simple, finite, connected and undirected graph of order n and size m

Total Global Dominator Coloring of Trees and Unicyclic Graphs

          A total global dominator coloring of a graph  is a proper vertex coloring of  with respect to which every vertex  in  dominates a color class, not containing  and does not dominate another color class. The minimum number of colors required in such a coloring of  is called the total global dominator chromatic number, denoted by . In this paper, the total global dominator chromatic number of trees and unicyclic graphs are explored

Average distance colouring of graphs

For a graph G with n vertices, average distance µ(G) is the ratio of sum of the lengths of the shortest paths between all pairs of vertices to the number of edges in a complete graph on n vertices. In this paper, we introduce average distance colouring and find the average distance colouring number of certain classes of graphs

New results on connected dominating structures in graphs

A set of vertices in a graph is a dominating set if every vertex not in the set is adjacent to at least one vertex in the set. A dominating structure is a subgraph induced by the dominating set. Connected domination is a type of domination where the dominating structure is connected. Clique domination is a type of domination in graphs where the dominating structure is a complete subgraph. The clique domination number of a graph G denoted by γk(G) is the minimum cardinality among all the clique dominating sets of G. We present few properties of graphs admitting dominating cliques along with bounds on clique domination number in terms of order and size of the graph. A necessary and sufficient condition for the existence of dominating clique in strong product of graphs is presented. A forbidden subgraph condition necessary to imply the existence of a connected dominating set of size four also is found

-energy of graphs

Given a graph G = (V, E), with respect to a vertex partition we associate a matrix called -matrix and define the -energy, E (G) as the sum of -eigenvalues of -matrix of G. Apart from studying some properties of -matrix, its eigenvalues and obtaining bounds of -energy, we explore the robust(shear) -energy which is the maximum(minimum) value of -energy for some families of graphs. Further, we derive explicit formulas for E (G) of few classes of graphs with different vertex partitions

Further results on color energy of graphs

Given a colored graph G, its color energy Ec(G) is defined as the sum of the absolute values of the eigenvalues of the color matrix of G. The concept of color energy was introduced by Adiga et al. [1]. In this article, we obtain some new bounds for the color energy of graphs and establish relationship between color energy Ec(G) and energy E(G) of a graph G. Further, we construct some new families of graphs in which one is non-co-spectral color-equienergetic with some families of graphs and another is color-hyperenergetic. Also we derive explicit formulas for their color energies