arXiv subjects
Publications and source records attributed to M. A. Seoud.
This manuscript introduces Diophantine labeling, a new way of labeling of the vertices for finite simple undirected graphs with some divisibility condition on the edges. Maximal graphs admitting Diophantine labeling are investigated and their number of edges are computed. Some number-theoretic techniques are used to characterize vertices of maximum degree and nonadjacent vertices. Some necessary and sufficient conditions for vertices of equal degrees are found.
A linear Diophantine equation $ax + by = n$ is solvable if and only if gcd$(a; b)$ divides $n$. A graph $G$ of order $n$ is called Diophantine if there exists a labeling function $f$ of vertices such that gcd$(f(u); f(v))$ divides $n$ for every two adjacent vertices $u; v$ in $G$. In this work, maximal Diophantine graphs on $n$ vertices, $D_n$, are defined, studied and generalized. The independence number, the number of vertices with full degree and the clique number of $D_n$ are computed. Each of these quantities is the basis of a necessary condition for the existence of such a labeling.
In this article, for an $\emph{MS}$- algebra and a fuzzy filter $χ$, the concept of extended fuzzy filter of $χ$ is presented, notated by $ Υ_{χ,W}$ with $W \subseteq \mathcal{L}$. The features of $Υ_{χ,W}$ are investigated. Furthermore, the strong fuzzy filter is introduced, donated by $Ω_{χ,W}$. Many properties are studied. Characterisation of both $ Υ_{χ,W}, Ω_{χ,W}$ are clarified by using the notion of dense elements with respect to a fuzzy filter. The homomorphisms of both $Υ_{χ,W}$ and $Ω_{χ,W}$ are triggered.
In this paper, we discuss difference labeling of some standard families of graphs. We prove that Star, Butterfly, Bistar, umbrella and Olive tree are difference graphs. We also introduce difference labelings for some snakes (double triangular snake, irregular triangular snake, alternate $C_n$ snake). Furthermore we introduce a corollary helps us to find a unique difference labeling for the complete graph $K_3$ and all forms of difference labeling for the Star graph. Also this corollary can be used to prove that the complete bipartite graph $K_{2,4}$ is not a difference graph but the proof is very lengthy.