A balanced coloring of a graph G means a triple {P1, P2,X} of mutually disjoint subsets of the vertex-set V (G) such that V (G) = P1⊎P2⊎X and |P1| = |P2|. A balanced decomposition associated with the balanced coloring V (G) = P1⊎P2⊎X of G is defined as a partition of V (G) = V1⊎· · ·⊎Vr (for some r) such that, for every i ∈ {1, · · · , r}, the subgraphG[Vi] of G is connected and |Vi∩P1| = |Vi∩P...

The balanced Hamiltonian cycle problem is a quiet new topic of graph theorem. Given a graph G = (V, E), whose edge set can be partitioned into k dimensions, for positive integer k and a Hamiltonian cycle C on G. The set of all i-dimensional edge of C, which is a subset by E(C), is denoted as Ei(C). If ||Ei(C)| |Ej(C)|| 1 for 1 i < j k, C is called a balanced Hamiltonian cycle. In this paper, th...

In this paper, we discuss balanced bipolar intuitionistic fuzzy graphs and study some of their properties 1.Introduction Graph theory is developed when Euler gave the solution to the famous Konigsberg bridge problem in 1736. Graph theory is very useful as a branch of combinatorics in the field of geometry, algebra, number theory, topology, operations research, optimization and computer science....

Let G = (X, Y; E) be a balanced bipartite graph of order 2n. The path-cover number pc(H) of a graph H is the minimum number of vertex-disjoint paths that use up all the vertices of H. S ⊆ V(G) is called a balanced set of G if |S ∩ X| = |S ∩ Y|. In this paper, we will give some sufficient conditions for a balanced bipartite graph G satisfying that for every balanced set S, there is a bi-cycle of...

We consider simple labeled graphs, with non-zero labels in a ring. If the adjacency matrix of a labeled graph is invertible, the inverse matrix is a (labeled) adjacency matrix of another graph, called the inverse of the original graph. If the labeling takes place in an ordered ring, then balanced inverses—those with positive products of labels along every cycle—are of interest. We introduce the...

A divisible design graph is a graph whose adjacency matrix is the incidence matrix of a divisible design. Divisible design graphs are a natural generalization of (v, k, λ)-graphs, and like (v, k, λ)-graphs they make a link between combinatorial design theory and algebraic graph theory. The study of divisible design graphs benefits from, and contributes to, both parts. Using information of the e...

Let G be a graph. The first Zagreb M1(G) of graph G is defined as: M1(G) = uV(G) deg(u)2. In this paper, we prove that each even number except 4 and 8 is a first Zagreb index of a caterpillar. Also, we show that the fist Zagreb index cannot be an odd number. Moreover, we obtain the fist Zagreb index of some graph operations.

the chromatic number of a graph g, denoted by χ(g), is the minimum number of colors such that g can be colored with these colors in such a way that no two adjacent vertices have the same color. a clique in a graph is a set of mutually adjacent vertices. the maximum size of a clique in a graph g is called the clique number of g. the turán graph tn(k) is a complete k-partite graph whose partition...