Tooth bleaching is becoming increasingly popular among patients and dentists since it is a relatively noninvasive approach for whitening and lightening teeth. Instruments and visual assessment with respect to commercial shade guides are currently used to evaluate tooth color. However, the association between these procedures is imprecise and the degree of color change after tooth bleaching is k...

let $p$ be a prime number and $n$ be a positive integer. the graph $g_p(n)$ is a graph with vertex set $[n]={1,2,ldots ,n}$, in which there is an arc from $u$ to $v$ if and only if $uneq v$ and $pnmid u+v$. in this paper it is shown that $g_p(n)$ is a perfect graph. in addition, an explicit formula for the chromatic number of such graph is given.

This note presents two results on real zeros of chromatic polynomials. The first result states that if G is a graph containing a q-tree as a spanning subgraph, then the chromatic polynomial P (G,λ) of G has no non-integer zeros in the interval (0, q). Sokal conjectured that for any graph G and any real λ > ∆(G), P (G,λ) > 0. Our second result confirms that it is true if ∆(G) ≥ bn/3c − 1, where ...

Given a minor-closed class of graphs G, what is the infimum of the non-trivial roots of the chromatic polynomial of G ∈ G? When G is the class of all graphs, the answer is known to be 32/27. We answer this question exactly for three minorclosed classes of graphs. Furthermore, we conjecture precisely when the value is larger than 32/27.

Given a graph we show how to construct a family of manifolds whose Euler characteristics are the values of the chromatic polynomial of the graph at various integers. The manifolds are simple generalisations of configuration spaces.

A class of graphs called generalized ladder graphs is defined. A sufficient condition for pairs of these graphs to be chromatically equivalent is proven. In addition a formula for the chromatic polynomial of a graph of this type is proven. Finally, the chromatic polynomials of special cases of these graphs are explicitly computed.

Suppose (P,-<) is a poset of size n and n: P-~ P is a permutation. We say that n has a drop at x if n(x)~x. Let fie(k) denote the number of n having k drops, 0 <~ k < n, and define the drop polynomial A p(2) by Further, define the incomparability graph I(P) to have vertex set P and edges 0" whenever i and j are incomparable in P, i.e., neither i-<j nor j< i holds. In this note we give a short p...

Let P(G;) denote the chromatic polynomial of a graph G. A graph G is chromatically unique if G ∼ = H for any graph H such that P(H;) = P(G;). This note corrects an error in the proof of the chromatic uniqueness of certain 2-connected graphs with n vertices and n + 3 edges.

The notion of the b-chromatic number of a graph attracted much research interests and recently a new concept, namely the b-chromatic sum of a graph, denoted by (G), has also been introduced. Motivated by the studies on b-chromatic sum of graphs, in this paper we introduce certain new parameters such as -chromatic sum, -chromatic sum, b-chromatic sum, -chromatic sum and -chromatic sum of graphs....

Sophisticated fuzzy rule systems are supposed to be flexible, complete, consistent and compact (FC). Flexibility, completeness and consistency are essential for fuzzy systems to exhibit an excellent performance and to have a clear physical meaning, while compactness is crucial when the number of the input variables increases. However, the completeness and consistency conditions are often violat...