We present a matching and LP based heuristic algorithm that decides graph non-Hamiltonicity. Each of the n! Hamilton cycles in a complete directed graph on n + 1 vertices corresponds with each of the n! n-permutation matrices P, such that pu,i = 1 if and only if the ith arc in a cycle enters vertex u, starting and ending at vertex n + 1. A graph instance (G) is initially coded as exclusion set ...

‎let $r$ be a commutative ring with non-zero identity. ‎we describe all $c_3$‎- ‎and $c_4$-free intersection graph of non-trivial ideals of $r$ as well as $c_n$-free intersection graph when $r$ is a reduced ring. ‎also, ‎we shall describe all complete, ‎regular and $n$-claw-free intersection graphs. ‎finally, ‎we shall prove that almost all artin rings $r$ have hamiltonian intersection graphs. ...

A graph is said to be symmetric if its automorphism group is transitive on its arcs. A complete classification is given of pentavalent symmetric graphs of order 24p for each prime p. It is shown that a connected pentavalent symmetric graph of order 24p exists if and only if p=2, 3, 5, 11 or 17, and up to isomorphism, there are only eleven such graphs.

‎Let $R$ be a commutative ring with non-zero identity. ‎We describe all $C_3$‎- ‎and $C_4$-free intersection graph of non-trivial ideals of $R$ as well as $C_n$-free intersection graph when $R$ is a reduced ring. ‎Also, ‎we shall describe all complete, ‎regular and $n$-claw-free intersection graphs. ‎Finally, ‎we shall prove that almost all Artin rings $R$ have Hamiltonian intersection graphs. ...

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].

Graph theory has an important role in the area of applications of networks and clustering‎. ‎In the case of dealing with uncertain data‎, ‎we must utilize ambiguous data such as fuzzy value‎, ‎fuzzy interval value or values of fuzzy number‎. ‎In this study‎, ‎values of fuzzy number were used‎. ‎Initially‎, ‎we utilized the fuzzy number value fuzzy relation and then proposed fuzzy number-value f...