Resistance distance and the normalized Laplacian spectrum

It is well known that the resistance distance between two arbitrary vertices in an electrical network can be obtained in terms of the eigenvalues and eigenvectors of the combinatorial Laplacian matrix associated with the network. By studying this matrix, people have proved many properties of resistance distances. But in recent years, the other kind of matrix, named the normalized Laplacian, which is consistent with the matrix in spectral geometry and random walks [Chung, F.R.K., Spectral Graph Theory, American Mathematical Society: Providence, RI, 1997], has engendered people's attention. For many people think the quantities based on this matrix may more faithfully reflect the structure and properties of a graph. In this paper, we not only show the resistance distance can be naturally expressed in terms of the normalized Laplacian eigenvalues and eigenvectors of G, but also introduce a new index which is closely related to the spectrum of the normalized Laplacian. Finally we find a non-trivial relation between the well-known Kirchhoff index and the new index.     

On fans in multigraphs

We introduce a unifying framework for studying edge‐coloring problems on multigraphs. This is defined in terms of a rooted directed multigraph , which is naturally associated to the set of fans based at a given vertex u in a multigraph G. We call the “Fan Digraph.” We show that fans in G based at u are in one‐to‐one correspondence with directed trails in starting at the root of . We state and prove a central theorem about the fan digraph, which embodies many edge‐coloring results and expresses them at a higher level of abstraction. Using this result, we derive short proofs of classical theorems. We conclude with a new, generalized version of Vizing's Adjacency Lemma for multigraphs, which is stronger than all those known to the author. © 2005 Wiley Periodicals, Inc. J Graph Theory 51: 301–318, 2006     

Completions of fans

In a finite-dimensional real vector space furnished with a rational structure with respect to a subfield of the field of real numbers, every (simplicial) rational semifan is contained in a complete (simplicial) rational semifan. In this paper this result is proved constructively on use of techniques from polyhedral geometry.     

Horizontal and vertical mutation fans

We introduce diagrams for m-cluster categories which we call "horizontal" and "vertical" mutation fans. These are analogous to the mutation fans(also known as "semi-invariant pictures" or "scattering diagrams") for the standard(m = 1) cluster case, which are dual to the poset of finitely generated torsion classes. The purpose of these diagrams is to visualize mutations and analogues of maximal green sequences in the m-cluster category with special emphasis on the c-vectors(the "brick" labels).     

Folded fans and string functions

Folded fans Fψ describe recursion properties of weights of integrable highest weight modules L ^u . Being considered simultaneously for the set of string functions s_s^u sigma_s^u belonging to a fundamental Weyl chamber and corresponding to the same congruence class, the system of recursion relations gives rise to an equation which connects the string functions and the power series depending on multiplicities of weights of a folded fan Fψ. We apply these equations to study properties of string functions s_s^u sigma_s^u associated with integrable modules for affine Lie algebras. New important relations for string functions are thus obtained. The set of folded fans provides a compact and effective tool to study them. Bibliography: 7 titles.     

Wheels within wheels

A simple means of representing the structure of all strongly connected directed graphs is developed and applied to a certain problem involving the construction of “toll booths”.     

Wave fans are special

It is shown that self-similar BV solutions of genuinely nonlinear strictly hyperbolic systems of conservation laws are special functions of bounded variation, with vanishing Cantor part.     

Secondary fans and tropical Severi varieties

This article studies the relationship between tropical Severi varieties and secondary fans. In the case when tropical Severi varieties are hypersurfaces this relationship is very well known; specifically, in this case, a tropical Severi variety of codimension 1 is a subfan of the corresponding secondary fan. It was expected for some time that this continues to hold more generally, but Katz found a counterexample in codimension 2, showing that this relationship is more subtle. The two main results in this paper are as follows. The first theorem finds a simple condition under which a tropical Severi variety cannot be a subfan of the corresponding secondary fan. The second theorem provides a partial converse, namely, we find conditions under which a cone of the secondary fan is fully contained in the tropical Severi variety. As a first application of these results, we also find a combinatorial formula for the tropical intersection multiplicities for secondary fans.     

On Ramsey numbers of fans

It is shown that r(F₂,F_n)=4n+1 for n≥2, and r(F_s,F_n)≤4n+2s for n≥s≥2.     

Lattice congruences, fans and Hopf algebras

We give a unified explanation of the geometric and algebraic properties of two well-known maps, one from permutations to triangulations, and another from permutations to subsets. Furthermore we give a broad generalization of the maps. Specifically, for any lattice congruence of the weak order on a Coxeter group we construct a complete fan of convex cones with strong properties relative to the corresponding lattice quotient of the weak order. We show that if a family of lattice congruences on the symmetric groups satisfies certain compatibility conditions then the family defines a sub Hopf algebra of the Malvenuto–Reutenauer Hopf algebra of permutations. Such a sub Hopf algebra has a basis which is described by a type of pattern avoidance. Applying these results, we build the Malvenuto–Reutenauer algebra as the limit of an infinite sequence of smaller algebras, where the second algebra in the sequence is the Hopf algebra of non-commutative symmetric functions. We also associate both a fan and a Hopf algebra to a set of permutations which appears to be equinumerous with the Baxter permutations.     

Tip-clearance and other three-dimensional effects in axial flow fans

Zusammenfassung Es wird die an einem Axialventilator durchgeführte Untersuchung der Auswirkung des Schaufelspaltes auf Laufradstr?mung und Leistung beschrieben. Das Laufrad, mit einem Durchmesser von 305 mm, war für ausgepr?gt dreidimensionale Str?mung entworfen. Die in solchen F?llen bestehenden Einschr?nkungen der Gültigkeit von zweidimensionalen Berechnungsunterlagen aus Windkanalversuchen werden besprochen. Es zeigte sich, dass die radial nach aussen gerichtete Querstr?mung durch Zentrifugalwirkung und die nach innen gerichtete Querstr?mung durch Vergr?sserung des Spaltspieles hervorgerufen wird. Vergr?sserung der Spaltweite beeinflusst die Str?mungsverh?ltnisse nicht nur in der N?he der Schaufelspitze, sondern über die ganze radiale Erstreckung der Schaufel. Dadurch wird die Druckziffer verkleinert, und die Abreissbedingungen ver?ndern sich. Unter diesen Verh?ltnissen ist es wesentlich, die ?nderung der Axialgeschwindigkeit im Laufrad korrekt in Rechnung zu setzen. Der wesentlichste Parameter scheint die Gr?ssec _2m /u zu sein. Dabei zeichnet sich bei gleichen Eintrittsverh?ltnissen, aber stark verschiedenen Querstr?mungen ein eindeutiger Zusammenhang zwischenc _2u /c _2m undc _2m /u für beliebige Schaufelgeometrien ab. Es mag daher von Vorteil sein, den aus Messungen am zweidimensionalen Windkanal erhaltenen Unterlagen in dieser Form den Vorzug vor den üblichen Einzelflügel- und Gitterparametern zu geben, die durch Querstr?mung merklich beeinflusst werden.      

The chromaticity of wheels

In this paper we prove that the wheel W_n+1 is chromatically unique if n is even. We also show that W₈ is not chromatically unique.     

Open selections on smooth fans

Given a dendroid X, an open selection is an open map such that s(A)∈A for every A∈C(X). We show that a smooth fan X admits an open selection if and only if X is locally connected.     

On hyperbolic virtual polytopes and hyperbolic fans

Hyperbolic virtual polytopes arose originally as polytopal versions of counterexamples to the following A.D.Alexandrov’s uniqueness conjecture: Let K ⊂ ℝ³ be a smooth convex body. If for a constant C, at every point of ∂K, we have R ₁ ≤ C ≤ R ₂ then K is a ball. (R ₁ and R ₂ stand for the principal curvature radii of ∂K.) This paper gives a new (in comparison with the previous construction by Y.Martinez-Maure and by G.Panina) series of counterexamples to the conjecture. In particular, a hyperbolic virtual polytope (and therefore, a hyperbolic hérisson) with odd an number of horns is constructed. Moreover, various properties of hyperbolic virtual polytopes and their fans are discussed.