The chromatic sum of a graph G, ∑ (G), is introduced in the dissertation of Kubicka [3]. It is defined as the smallest possible total over all vertices that can occur among all colorings of G using natural numbers for the colors. It is known that computing the chromatic sum of an arbitrary graph is an NP-complete problem. The vertex-strength of the graph G, denoted by s(G), is the smallest inte...

In this paper we introduced and studies the concepts of independent domination bipolar fuzzy graph G, (?i(G)). And chromatic number ? (G) .We investigated relationship ?i with other known parameters. Finally give for same standard

the chromatic dispersion curve of the fundamental mode in small core microstructured fibres (SCMF) is both calculated using a Finite Element Method (FEM) and measured with a low coherence interferometric method. The great sensitivity of the chromatic dispersion to variations of the geometrical parameters of SCMFs (the pitch  and the diameter d) is pointed out. An excellent agreement is obtaine...

In a graph G, a vertex dominates itself and its neighbors. A subset S of V is called a dominating set in G if every vertex in V-S is adjacent to at least one vertex in S. The minimum cardinality taken over all, the minimal double dominating set which is called Fuzzy Double Domination Number and which is denoted as ) (G fdd  . A set V S  is called a Triple dominating set of a graph G if every ...

In the main this paper introduces the concept of chromatic harmonic polynomials denoted, $H^chi(G,x)$ and chromatic harmonic indices denoted, $H^chi(G)$ of a graph $G$. The new concept is then applied to finding explicit formula for the minimum (maximum) chromatic harmonic polynomials and the minimum (maximum) chromatic harmonic index of certain graphs. It is also applied to split graphs and ce...

Bipolar fuzzy graph (BFG) coloring techniques are used to solve many complex real world problems. The chromatic number of complement BFG is obtained and compared with the corresponding BFGs. This paper an attempt define in a based on strong edges. complete BF tree obtained.

Objectives are usually imprecise (fuzzy). In many real-life situations, our objectives are imprecise (fuzzy). A company may want to have a good growth rate, an excellent level of customer satisfaction, etc. A university program may seek excellent quality of graduates, steady growth of the program, more good students and faculty in the program, etc. Many such statements use imprecise (fuzzy) wor...