For a graph $G$, let $P(G,lambda)$ denote the chromatic polynomial of $G$. Two graphs $G$ and $H$ are chromatically equivalent if they share the same chromatic polynomial. A graph $G$ is chromatically unique if any graph chromatically equivalent to $G$ is isomorphic to $G$. A $K_4$-homeomorph is a subdivision of the complete graph $K_4$. In this paper, we determine a family of chromatically uni...

For a graphG, let P(G, λ)be its chromatic polynomial. TwographsG andH are chromatically equivalent, denoted G ∼ H , if P(G, λ) = P(H, λ). A graph G is chromatically unique if P(H, λ) = P(G, λ) implies that H ∼= G. In this paper, we shall determine all chromatic equivalence classes of 2-connected (n, n+ 4)-graphs with three triangles and one induced 4-cycle, under the equivalence relation ‘ ∼’. ...

Let P(G) denote the chromatic polynomial of a graph G. Two graphs G and H are chromatically equivalent, written G∼H , if P(G) = P(H). A graph G is chromatically unique if for any graph H , G∼H implies that G is isomorphic with H . In this paper, we give the necessary and su:cient conditions for a family of generalized polygon trees to be chromatically unique. c © 2001 Elsevier Science B.V. All ...

For any positive integer n, let Gn denote the set of simple graphs of order n. For any graph G in Gn, let P(G; ) denote its chromatic polynomial. In this paper, we -rst show that if G ∈Gn and (G)6 n− 3, then P(G; ) is zero-free in the interval (n − 4 + =6 − 2= ;+∞), where = (108 + 12√93)1=3 and =6 − 2= (=0:682327804 : : :) is the only real root of x + x − 1; we proceed to prove that whenever n ...

Let ) , ( λ χ G denotes the number of proper vertex colourings of with at most G λ colours. G. Birkhoff , observed in 1912 that ) , ( λ χ G is, for a fixed graph , a polynomial in G λ , which is now called the chromatic polynomial of . More precisely, let G be a simple graph and G N ∈ λ . A mapping } , {1,2, ) ( : λ K → G V f is called a λ -colouring of if whenever the vertices and are adjacent...

Recently, we developed a method for optimally estimating focus error given a set of natural scenes, a waveoptics model of the lens system, a sensor array, and a specification of measurement noise. The method is based on first principles and can be tailored to any vision system for which these properties can be characterized. Here, the method is used to estimate defocus in local areas of images ...

Although there are many excellent ways presenting the principle of the classical calculus, the novel presentations probably leads most naturally to the development of the non-Newtonian calculus. The important point to note is that the non-Newtonian calculus is a self-contained system independent of any other system of calculus. Since this self-contained work is intended for a wide audience, inc...

Localization based super-resolution microscopy techniques require precise drift correction methods because the achieved spatial resolution is close to both the mechanical and optical performance limits of modern light microscopes. Multi-color imaging methods require corrections in addition to those dealing with drift due to the static, but spatially-dependent, chromatic offset between images. W...

Fuzzy metagraph is an emerging technique used in the design of many information processing systems like transaction processing systems, decision support systems, and workflow systems. Very often, even a carefully chosen graph data structure could be improvised to provide more efficiency in terms of time complexity or space complexity or both. In this paper, a well-designed fuzzy metagraph is pr...

A new complex neuro-fuzzy self-learning approach to the problem of function approximation is proposed, where complex fuzzy sets are used to design a complex neuro-fuzzy system as the function approximator. Particle swarm optimization (PSO) algorithm and recursive least square estimator (RLSE) algorithm are used in hybrid way to adjust the free parameters of the proposed complex neuro-fuzzy syst...