Cycle multiplicity of a graph G is the maximum number edge disjoint cycles in G. In this paper, we determine cycle and then obtain formula total complete bipartite graph, generalizes result for, which given by M.M. Akbar Ali [1].

a watching system in a graph $g=(v, e)$ is a set $w={omega_{1}, omega_{2}, cdots, omega_{k}}$, where $omega_{i}=(v_{i}, z_{i}), v_{i}in v$ and $z_{i}$ is a subset of closed neighborhood of $v_{i}$ such that the sets $l_{w}(v)={omega_{i}: vin omega_{i}}$ are non-empty and distinct, for any $vin v$. in this paper, we study the watching systems of line graph $k_{n}$ which is called triangular grap...

In this paper, we prove that every semi-Cayley graph over a group G is quasi-abelian if and only if G is abelian.

A total coloring of a graph G is a coloring of its vertices and edges such that no adjacent vertices, edges, and no incident vertices and edges obtain the same color. An interval total t-coloring of a graph G is a total coloring of G with colors 1, . . . , t such that all colors are used, and the edges incident to each vertex v together with v are colored by dG(v) + 1 consecutive colors, where ...

In this paper a graph is considered as finite, undirected, with single lines and no loops. The arboricity of a graph G, denoted by Y(G), is the least number of line-disjoint spanning forests into which G can be partitioned. {x} is the least integer not less than X, and A(G) is the maximum degree among the points of G. V(G), E(G), and ISI denote the point set of G, the line set of G, and the num...