Representation of an algebra associated with the Dynkin graph $$\tilde E_7$$

We describe the structure of pairs of self-adjoint operators A and B whose spectra belong to the set {±1/2, ±3/2} and for which (A+B)²=I. Such pairs of operators determine a representation of a *-algebra associated with the extended Dynkin graph .__________Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 9, pp. 1193–1202, September, 2004.This revised version was published online in May 2005 with a corrected article title.     

A property of the colored complete graph

If the lines of the complete graph K_n are colored so that no point is on more than $\text{1}\text{7}\text{(n?1)}$ lines of the same color or so that each point lies on more than $\text{1}\text{7}\text{(5n+8)}$ lines of different colors, then K_n contains a cycle of length n with adjacent lines having different colors.     

The Terwilliger algebra of the incidence graph of the Hamming graph

Let \(\varGamma \) be a distance-semiregular graph on Y, and let \(D^Y\) be the diameter of \(\varGamma \) on Y. Let \(\varDelta \) be the halved graph of \(\varGamma \) on Y. Fix \(x \in Y\). Let T and \(T'\) be the Terwilliger algebras of \(\varGamma \) and \(\varDelta \) with respect to x, respectively. Assume, for an integer i with \(1 \le 2i \le D^Y\) and for \(y,z \in \varGamma _{2i}(x)\) with \(\partial _{\varGamma }(y,z)=2\), the numbers \(|\varGamma _{2i-1}(x) \cap \varGamma (y) \cap \varGamma (z)|\) and \(|\varGamma _{2i+1}(x) \cap \varGamma (y) \cap \varGamma (z)|\) depend only on i and do not depend on the choice of y, z. The first goal in this paper is to show the relations between T-modules of \(\varGamma \) and \(T'\)-modules of \(\varDelta \). Assume \(\varGamma \) is the incidence graph of the Hamming graph H(D, n) on the vertex set Y and the set \({\mathcal {C}}\) of all maximal cliques. Then, \(\varGamma \) satisfies above assumption and \(\varDelta \) is isomorphic to H(D, n). The second goal is to determine the irreducible T-modules of \(\varGamma \). For each irreducible T-module W, we give a basis for W the action of the adjacency matrix on this basis and we calculate the multiplicity of W.     

A contact process with a semi-infected state on the complete graph

In this paper, we are concerned with a contact process with a semi-infected state on the complete graph C_n with n vertices. Our model is a special case of a general model introduced by Schinazi in 2003. In our model, each vertex is in one of three states, namely, “healthy,” “semi-infected,” or “fully-infected.” Only fully-infected vertices can infect others. A healthy vertex becomes semi-infected when being infected while a semi-infected vertex becomes fully-infected when being further infected. Each (semi- and fully-) infected vertex becomes healthy at constant rate. Our main result shows a phase transition for the waiting time until extinction of the fully-infected vertices. Conditioned on all the vertices are fully-infected when t = 0, we show that fully-infected vertices survive for exp?{O(n)} units of time when the infection rate λ > 4 while they die out in O(log?n) units of time when λ < 4.     

A linear algebra setting for the rota-mullin theory of polynomials of binomial type

The linear algebra and combinatorial aspects of the Rota-Mullin theory of polynomials of binomial type are separated and the former is developed in terms of shift operators on infinite dimensional vector spaces with a view towards application in the calculus of finite differences.     

On the line graph of the complete tripartite graph

It is shown that the line graph of the complete tripartite graphK_n,n,n is characterized by the spectrum of its adjacency matrix.     

Packings of the complete directed graph with m-circuits

A packing of the complete directed symmetric graph DKv with m-circuits, denoted by(v,m)-DCP, is defined to he a family of are-disjoint m-circuits of DK, such that any one arc of DKv occurs in at most one m circuit. The packing number P(v,m) is the maximum number of m-circults in such a packing. The packing problem is to determine the value P(v,m) for everyinteger v ≥ m. In this paper, the problem is reduced to the case m 6 ≤v≤2m-[(4m-3的平方极) 1/2],for any fixed even integer m≥4,In particular,the values of P（v,m） are completely determined for m=12,14 and 16.     

A new Lie algebra along with its induced Lie algebra and associated with applications

A 3 × 3 Lie algebra H is introduced whose induced Lie algebra by decomposition and linear combinations is obtained, which may reduce to the Lie algebra given by AP Fordy and J Gibbons. By employing the induced Lie algebra and the zero curvature equation, a kind of enlarged Boussinesq soliton hierarchy is produced. Again making use of a subalgebra of the induced Lie algebra leads to the well-known KdV hierarchy whose expanding integrable system is also worked out. As an applied example of the Lie algebra H, we obtain a new integrable coupling of the well-known AKNS hierarchy.     

On the extender algebra being complete

We show that a Woodin cardinal is necessary for the extender algebra to be complete. Our proof is relatively simple and does not use fine structure. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A specialization of prinjective Ringel-Hall algebra and the associated Lie algebra

In the present paper we describe a specialization of prinjective Ringel-Hall algebra to 1, for prinjective modules over incidence algebras of posets of finite prinjective type, by generators and relations. This gives us a generalisation of Serre relations for semisimple Lie algebras. Connections of prinjective Ringel-Hall algebras with classical Lie algebras are also discussed.     

A C5-decomposition of the λ-fold line graph of the complete graph

In 1996, Cox and Rodger [Cycle systems of the line graph of the complete graph, J. Graph Theory 21 (1996) 173–182] raised the following question: For what values of $m$ and $n$ does there exist an $m$-cycle decomposition of $L\left(K_n\right)?$ In this paper, the above question is answered for $m=5.$ In fact, it is shown that $L\left(K_n\right)\left(\lambda \right),$ the $\lambda$-fold line graph of the complete graph $K_n,$ has a $C_5$-decomposition if and only if $5\mid \lambda \left(\genfrac{}{}{0ex}{}{n}{2}\right)(n?2)$ and $n\ge 4.$     

A model of ZF with an infinite free complete Boolean algebra

By a theorem of Gaifman and Hales no model of ZF+AC (Zermelo-Fraenkel set theory plus the axiom of choice) contains an infinite free complete Boolean algebra. We construct a model of ZF in which an infinite free c.B.a. exists.     

A semi-finite algebra associated to a subfactor planar algebra

We canonically associate to any planar algebra two type II_∞ factors M_±. The subfactors constructed previously by the authors in Guionnet et al. (2010) [6] are isomorphic to compressions of M_± to finite projections. We show that each M_± is isomorphic to an amalgamated free product of type I von Neumann algebras with amalgamation over a fixed discrete type I von Neumann subalgebra. In the finite-depth case, existing results in the literature imply that M₊≅M₋ is the amplification a free group factor on a finite number of generators. As an application, we show that the factors Mj constructed in Guionnet et al. (in press) [6] are isomorphic to interpolated free group factors L(F(rj)), rj=1+2δ−2j(δ−1)I, where δ² is the index of the planar algebra and I is its global index. Other applications include computations of laws of Jones-Wenzl projections.     

A discrete Fourier transform associated with the affine Hecke algebra

We introduce an explicit representation of the double affine Hecke algebra (of type _A1$A_1$) at q=1$q=1$ that gives rise to a periodic counterpart of a well-known Fourier transform associated with the affine Hecke algebra.     

On the total variation distance between the binomial random graph and the random intersection graph

When each vertex is assigned a set, the intersection graph generated by the sets is the graph in which two distinct vertices are joined by an edge if and only if their assigned sets have a nonempty intersection. An interval graph is an intersection graph generated by intervals in the real line. A chordal graph can be considered as an intersection graph generated by subtrees of a tree. In 1999, Karoński, Scheinerman, and Singer‐Cohen introduced a random intersection graph by taking randomly assigned sets. The random intersection graph has n vertices and sets assigned to the vertices are chosen to be i.i.d. random subsets of a fixed set M of size m where each element of M belongs to each random subset with probability p, independently of all other elements in M. In 2000, Fill, Scheinerman, and Singer‐Cohen showed that the total variation distance between the random graph and the Erdös‐Rényi graph tends to 0 for any if , where is chosen so that the expected numbers of edges in the two graphs are the same. In this paper, it is proved that the total variation distance still tends to 0 for any whenever .     

Cycle systems of the line graph of the complete graph

For all m (identically equal to) 0 (mod 4), for all n (identically equal to) 0 or 2 (mod m), and for all n (identically equal to) 1 (mod 2m) we find an m-cycle decomposition of the line graph of the complete graph K_n. In particular, this solves the existence problem when m is a power of two. © 1996 John Wiley & Sons, Inc.     

Integrality of the symmetric algebra of graph ideals

We consider edge ideals associated to some classes of simple graphs and study the projective dimension and the integrality of the symmetric algebra for them. We also analyze criteria for torsion freeness of the symmetric powers and determine conditions of acyclicity of the Z-complex of these graph ideals.