Results 81 to 90 of about 1,320 (180)
A chromatic partition polynomial
Discrete Mathematics, 1998 Let \(|\pi|\) denote the number of blocks in a partition \(\pi\) of \([n]= \{1,2, \dots, n\}\). Let \(p=a_1 a_2 \dots a_n\) be a permutation of \([n]\). A descent block of \(p\) is a maximal decreasing continuance subword \(a_ia_{i+1} \dots a_j\) of \(p\). The \(n\)th Eulerian polynomial \(A_n(t)\) (for \(n=0,1,2,\dots)\) is defined by \[ \sum_{k\geq 0}
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Advanced Science, Volume 13, Issue 48, 28 August 2026.ABSTRACT The shift from point‐based thermal sensing to filter‐free thermal imaging requires luminescent thermometers that exhibit pronounced and thermally driven spectral changes within spectral regions matching the sensitivity profiles of the red, green and blue (RGB) channels of a digital camera.
Y. Abe +3 more
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Advanced Science, Volume 13, Issue 46, 17 August 2026.Task‐adaptive programmable optics enables label‐free virtual staining through optical‐attention‐guided acquisition and reconstruction. By optimizing wavelength, illumination angle, exposure time, and imaging depth, the framework learns task‐relevant optical measurements, generating clinically interpretable virtual stains with improved fidelity, non ...
Tianyue He +13 more
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A Note on a Broken-Cycle Theorem for Hypergraphs
Discussiones Mathematicae Graph Theory, 2014 Whitney’s Broken-cycle Theorem states the chromatic polynomial of a graph as a sum over special edge subsets.
Trinks Martin
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Transactions of the American Mathematical Society, 1946 Birkhoff, George D., Lewis, D. C.
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Chromatic Polynomials Of Some (m,l)-Hyperwheels [PDF]
Computer Science Journal of Moldova, 2014 In this paper, using a standard method of computing the chromatic polynomial of hypergraphs, we introduce a new reduction theorem which allows us to find explicit formulae for the chromatic polynomials of some (complete) non-uniform $(m,l)-$hyperwheels ...
Julian A. Allagan
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Chromatic polynomials of generalized trees
Discrete Mathematics, 1988 This short note surveys some recent results on chromatic polynomials of graphs built up in a treelike manner of q-cliques (yieldig q-trees) or of n-gons (n-gon-trees).
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Proof of a Chromatic Polynomial Conjecture
Journal of Combinatorial Theory, Series B, 2000 Let \(P(G,\lambda)\) be the chromatic polynomial of a graph \(G\) (i.e., the number of mappings \(f\) from the vertex set of \(G\) to \(\{1,2,\dots, \lambda\}\) such that \(f(x)\neq f(y)\) whenever \(x\) and \(y\) are adjacent vertices in \(G\) if \(\lambda\) is a positive integer).
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Hypergraphs with Pendant Paths are not Chromatically Unique
Discussiones Mathematicae Graph Theory, 2014 In this note it is shown that every hypergraph containing a pendant path of length at least 2 is not chromatically unique. The same conclusion holds for h-uniform r-quasi linear 3-cycle if r ≥ 2.
Tomescu Ioan
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A generalization of the chromatic polynomial of a cycle [PDF]
Computer Science Journal of Moldova, 2005 We prove that if an edge of a cycle on vertices is extended by adding vertices, then the the chromatic polynomial of such generalized cycle is: $$P(H_k,\lambda)=(\lambda-1)^n\sum_{i=0}^k \lambda^i+(-1)^n(\lambda-1).$$
Julian A. Allagan
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