Throughout this paper, R will denote a commutative ring with identity and M is a unitary R- module and Z will denote the ring of integers. We introduce the graph Ω(M) of module M with the set of vertices contain all nontrivial non-essential submodules of M. We investigate the interplay between graph-theoretic properties of Ω(M) and algebraic properties of M. Also, we assign the values of natura...

A graph G is a chordal-k-generalized split graph if G is chordal and there is a clique Q in G such that every connected component in G[V \ Q] has at most k vertices. Thus, chordal-1-generalized split graphs are exactly the split graphs. We characterize chordal-k-generalized split graphs by forbidden induced subgraphs. Moreover, we characterize a very special case of chordal-2-generalized split ...

A graph G = (V, E) is called a split graph if there exists a partition V = I ∪K such that the subgraphs G[I ] and G[K] of G induced by I and K are empty and complete graphs, respectively. In 1980, Burkard and Hammer gave a necessary condition for a split graph G with |I | < |K| to be hamiltonian. We will call a split graph G with |I | < |K| satisfying this condition a Burkard-Hammer graph. Furt...

Given a connected graph G and a terminal set R ⊆ V (G), Steiner tree asks for a tree that includes all of R with at most r edges for some integer r ≥ 0. It is known from [ND12,Garey et. al [1]] that Steiner tree is NP-complete in general graphs. Split graph is a graph which can be partitioned into a clique and an independent set. K. White et. al [2] has established that Steiner tree in split gr...

A rainbow colouring of a connected graph is a colouring of the edges of the graph, such that every pair of vertices is connected by at least one path in which no two edges are coloured the same. Such a colouring using minimum possible number of colours is called an optimal rainbow colouring, and the minimum number of colours required is called the rainbow connection number of the graph. A Chord...

It is known that determination problem of the bandwidth for an arbitrary graph is NP-complete. This paper establishes the bandwidth for the composition of a complete bipartite graph with other graphs; the strong product of a complete bipartite graph with a complete graph, a path, a cycle; and the tensor product of a complete graph with a complete graph.

LetKn1,n2,...,np denote the complete p-partite graph, p > 1, on n = n1+n2+ · · ·+np vertices and let n1 ≥ n2 ≥ · · · ≥ np > 0. We show that for a fixed value of n, both the spectral radius and the energy of complete p-partite graphs are minimal for complete split graph CS(n, p− 1) and are maximal for Turán graph T (n, p).

Finding the pebbling number of a graph is harder than NP-complete (Π 2 complete, to be precise). However, for many families of graphs there are formulas or polynomial algorithms for computing pebbling numbers; for example, complete graphs, products of paths (including cubes), trees, cycles, diameter two graphs, and more. Moreover, graphs having minimum pebbling number are called Class 0, and ma...

The Classification Problem is the problem of deciding whether a simple graph has chromatic index equals to ∆ or ∆+1, where ∆ is the maximum degree of the graph. It is known that to decide if a graph has chromatic index equals to ∆ is NP-complete. A split graph is a graph whose vertex set admits a partition into a stable set and a clique. The chromatic indexes for some subsets of split graphs, s...