Further results on graph equations for line graphs and n-th power graphs

We present solutions of seven graph equations involving the line graph, complement and n-th power operations. One such equation $\text{L(G)}{}^{\mathrm{n}}\text{=}\text{G}\text{?}$ generalizes a result of M. Aigner. In addition, some comments are made about graphs satisfying $\text{G}{}^{\mathrm{n}}\text{=}\text{G}\text{?}$.     

Graphes orientes indécomposables en circuits hamiltoniens

Let G be a cubic graph of order 2n consisting of a cycle plus a perfect matching and let $\text{G}{}^1$ be the symmetric digraph obtained from G by replacing each edge of G by two opposite arcs. In this paper we study when $\text{G}{}^1$ can be decomposed into three hamiltonian circuits and in particular we prove that such a decomposition is impossible if n is even.     

Elementary proof of the transformation formula for Lambert series involving generalized Dedekind sums

An elementary proof is given of the author's transformation formula for the Lambert series $\text{G}{}_{\mathrm{p}}\text{(x) =}\text{Σ}{}_{\mathrm{n?1}}{}^{\mathrm{<mtext>\infty </mtext>}}\text{n}{}^{\mathrm{?p}}\text{x}{}^{\mathrm{n}}\text{(1?x}{}^{\mathrm{n}}\text{)}$ relating G_p(e^2πiτ) to G_p(e^2πiAτ), where p > 1 is an odd integer and $\text{Aτ =}\text{(aτ + b)}\text{(cτ + d)}$ is a general modular substitution. The method extends Sczech's argument for treating Dedekind's function $\text{log}\text{η(τ) =}\text{πiτ}\text{12}\text{? G}{}_1\text{(e}{}^{\mathrm{2\pi i\tau }}\text{)}$, and uses Carlitz's formula expressing generalized Dedekind sums in terms of Eulerian functions.     

The asymptotic eigenvalue-distribution for a certain class of random matrices

For an n × n Hermitean matrix A with eigenvalues λ₁, …, λ_n the eigenvalue-distribution is defined by $\text{G(x, A) :=}\text{1}\text{n}$ · number {λ_i: λ_i ? x} for all real x. Let A_n for n = 1, 2, … be an n × n matrix, whose entries a_ik are for i, k = 1, …, n independent complex random variables on a probability space (Ω, $\text{R}$, p) with the same distribution F_a. Suppose that all moments $\text{E}$ | a | ^k, k = 1, 2, … are finite, $\text{E}$a=0 and $\text{E}$ | a | ². Let

$\text{M(A)=}\text{∑}\text{σ=1}\text{s}\text{θ}{}_{\mathrm{\sigma }}\text{P}{}_{\mathrm{\sigma }}\text{(A,A}{}^1\text{)}$

with complex numbers θ_σ and finite products P_σ of factors A and $\text{A}{}^1$ (= Hermitean conjugate) be a function which assigns to each matrix A an Hermitean matrix M(A). The following limit theorem is proved: There exists a distribution function G₀(x) = G₁x) + G₂(x), where G₁ is a step function and G₂ is absolutely continuous, such that with probability $\text{1 G(x, M(}\text{A}{}_{\mathrm{n}}\text{n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{))}$ converges to G₀(x) as n → ∞ for all continuity points x of G₀. The density g of G₂ vanishes outside a finite interval. There are only finitely many jumps of G₁. Both, G₁ and G₂, can explicitly be expressed by means of a certain algebraic function f, which is determined by equations, which can easily be derived from the special form of M(A). This result is analogous to Wigner's semicircle theorem for symmetric random matrices (E. P. Wigner, Random matrices in physics, SIAM Review9 (1967), 1–23). The examples $\text{A}{}_{\mathrm{r}}\text{A}{}^{\mathrm{1r}}$, $\text{A}{}^{\mathrm{r}}\text{+ A}{}^{\mathrm{1r}}$, $\text{A}{}^{\mathrm{r}}\text{A}{}^{\mathrm{1r}}\text{± A}{}^{\mathrm{1r}}\text{A}{}^{\mathrm{r}}$, r = 1, 2, …, are discussed in more detail. Some inequalities for random matrices are derived. It turns out that with probability 1 the sharpened form

$\text{lim sup}\text{n→∞}\text{∑}\text{i=1}{}^{\mathrm{n}}\text{|γ}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{|}{}^2\text{?6A}{}_{\mathrm{n}}\text{6}{}^2\text{? 0.8228…}$

of Schur's inequality for the eigenvalues λ_i⁽ⁿ⁾ of A_n holds. Consequently random matrices do not tend to be normal matrices for large n.     

The KMS condition and spectral passivity in group duality

Let (A, G, α) be a C¹-dynamical system, where G is abelian, and let φ be an invariant state. Suppose that there is a neighbourhood Ω of the identity in $\text{G}\text{?}$ and a finite constant κ such that $\text{Π}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{φ(x}{}_{\mathrm{i}}{}^1\text{x}{}_{\mathrm{i}}\text{) ? κ}\text{Π}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{φ(x}{}_{\mathrm{i}}\text{x}{}_{\mathrm{i}}{}^1\text{)}$ whenever x_i lies in a spectral subspace $\text{R}{}^{\mathrm{\alpha }}\text{(Ω}{}_{\mathrm{i}}\text{)}$, where $\text{Ω}{}_1\text{+ … + Ω}{}_{\mathrm{n}}\text{? Ω}$. This condition of complete spectral passivity, together with self-adjointness of the left kernel of φ, ensures that φ satisfies the KMS condition for some one-parameter subgroup of G.     

An infinite class of generalized room squares

A generalized Room square $\text{G}$ of order n and degree k is an ${}^{\mathrm{n?1}}{}^{\mathrm{k?1}}\text{×}{}^{\mathrm{n?1}}{}^{\mathrm{k?1}}$ array, each cell of which is either empty or contains an unordered k-tuple of a set S, |S| = n, such that each row and each column of the array contains each element of S exactly once and $\text{G}$ contains each unordered k-tuple of S exactly once. Using a class of Steiner systems and a generalized Room square of order 18 and degree 3 constructed by ad hoc methods, an infinite class of degree 3 squares is constructed.     

Factorizing the complete graph into factors with large star number

The graph G has star number n if any n vertices of G belong to a subgraph which is a star. Let f(n, k) be the smallest number m such that the complete graph on m vertices can be factorized into k factors with star number n. In the present paper we prove that $\text{c}{}_1{}^{\mathrm{n}}\text{k ≤ f(n, k) < c}{}_2{}^{\mathrm{n}}\text{k}$.     

A multiplication of distributions

If Ω denotes an open subset of $\text{R}$ⁿ (n = 1, 2,…), we define an algebra $\text{g}$ (Ω) which contains the space $\text{D}$′(Ω) of all distributions on Ω and such that $\text{C}{}^{\mathrm{\infty }}\text{(Ω)}$ is a subalgebra of $\text{G}$ (Ω). The elements of $\text{G}$ (Ω) may be considered as “generalized functions” on Ω and they admit partial derivatives at any order that generalize exactly the derivation of distributions. The multiplication in $\text{G}$(Ω) gives therefore a natural meaning to any product of distributions, and we explain how these results agree with remarks of Schwartz on difficulties concerning a multiplication of distributions. More generally if q = 1, 2,…, and $\text{?∈O}{}_{\mathrm{M}}\text{(}\text{R}{}^{\mathrm{2q}}\text{)}$—a classical Schwartz notation—for any G₁,…,G_q∈G(σ), we define naturally an element $\text{?G}{}_1\text{,…,G}{}_{\mathrm{q}}\text{∈G(σ)}$. These results are applied to some differential equations and extended to the vector valued case, which allows the multiplication of vector valued distributions of physics.     

Sufficiency and invariance

Suppose that a statistical decision problem is invariant under a group of transformations g?G. T (X) is equivariant if there exists $\text{g}{}^1\text{? G}{}^1$ such that $\text{T(g(X)) = g}{}^1\text{(T((X))}$. We show that the minimal sufficient statistic is equivalent and that if T(X) is an equivariant sufficient statistics and d(X) is invariant under G, then $\text{d}{}^1\text{(T) = Ed(X)∥T}$ is invariant under $\text{G}{}^1$.     

The phase transition in a random hypergraph

We show that in the evolution of the random d-uniform hypergraph $\text{G}{}^{\mathrm{d}}\text{(n,M)}$ the phase transition occurs when M=n/d(d−1)+O(n^2/3). We also prove local limit theorems for the distribution of the size of the largest component of $\text{G}{}^{\mathrm{d}}\text{(n,M)}$ in the subcritical and in the early supercritical phase.     

Vertex degrees of planar graphs

Let G be a planar graph having n vertices with vertex degrees d₁, d₂,…,d_n. It is shown that $\text{Σ}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{d}{}_{\mathrm{i}}{}^2\text{≤ 2n}{}^2\text{+ O(n)}$. The main term in this upper bound is best possible.     

Cycles of even length in graphs

In this paper we solve a conjecture of P. Erdös by showing that if a graph Gⁿ has n vertices and at least $\text{100kn}{}^{\mathrm{<mtext>1+</mtext><mtext>1</mtext><mtext>k</mtext>}}$ edges, then G contains a cycle C^2l of length 2l for every integer $\text{l ∈ [k, kn}{}^{\mathrm{<mtext>1</mtext><mtext>k</mtext>}}\text{]}$. Apart from the value of the constant this result is best possible. It is obtained from a more general theorem which also yields corresponding results in the case where Gⁿ has only cn(log n)^α edges (α ≥ 1).     

Positive definite distributions on K\GK

Let G be a semisimple noncompact Lie group with finite center and let K be a maximal compact subgroup. Then W. H. Barker has shown that if T is a positive definite distribution on G, then T extends to Harish-Chandra's Schwartz space $\text{C}$¹(G). We show that the corresponding property is no longer true for the space of double cosets $\text{K}\text{G}\text{K}$. If G is of real-rank 1, we construct liner functionals $\text{T}{}_{\mathrm{p}}\text{? (C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{))′}$ for each p, 0 < p ? 2, such that $\text{T}{}_{\mathrm{p}}\text{(f 1 f}{}^1\text{) ? 0, ?f ? C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{)}$ but T_p does not extend to a continuous functional on $\text{C}{}^{\mathrm{p}}\text{(K}\text{G}\text{K}\text{)}$. In particular, if p ? 1, T_v does not extend to a continuous functional on $\text{C}{}^1\text{(K}\text{G}\text{K}\text{)}$. We use this to answer a question (in the negative) raised by Barker whether for a K-bi-invariant distribution T on G to be positive definite it is enough to verify that $\text{T(f 1 f}{}^1\text{) ? 0, ?f ? C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{)}$. The main tool used is a theorem of Trombi-Varadarajan.     

Bordism groups and shape theory

The THOM isomorphism Theorem (1) allows an immediate extension to the strong shape category of compacta: The shape homology $\text{TBG}{}_{\mathrm{n}}\text{(X)}$ with coefficient in a THOM spectrum turns out to be isomorphic to the bordism group $\text{Ω}{}^{\mathrm{G}}{}_{\mathrm{n}}\text{(X)}$ which is defined like but with strong shape morphisms replacing continuous mappings (Theorem 1.2).     

A theoretical analysis of backtracking in the graph coloring problem

The graph coloring problem is: Given a positive integer K and a graph G. Can the vertices of G be properly colored in K colors? The problem is NP-complete. The average behavior of the simplest backtrack algorithm for this problem is studied. Average run time over all graphs is known to be bounded. Average run time over all graphs with n vertices and q edges behaves like $\text{exp}\text{(}\text{Cn}{}^2\text{q}\text{)}$. It is shown that similar results hold for all higher moments of the run time distribution. For all graphs and for graphs where lim $\text{n}{}^2\text{q}$ exists, the run time has a limiting disibution as n → ∞.     

On the asymptotic number of tournament score sequences

An n-tournament is a complete labelled digraph on n vertices without loops or multiple arcs. A tournament's score sequence is the sequence of the out-degrees of its vertices arranged in nondecreasing order. The number S_n of distinct score sequences arising from all possible n-tournaments, as well as certain generalizations are investigated. A lower bound of the form

$\text{S}{}_{\mathrm{n}}\text{>}\text{C}{}_1\text{4}{}^{\mathrm{n}}\text{n}{}^{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}$

(C₁ a constant) and an upper bound of the form $\text{S}{}_{\mathrm{n}}\text{<}\text{C}{}_2\text{4}{}^{\mathrm{n}}\text{n}{}^2$ are proved. A q-extension of the Catalan numbers

$\text{c}{}_1\text{(q)=1}\text{and}\text{c}{}_{\mathrm{n}}\text{(q)=}\text{∑}\text{i?1}\text{n?1}\text{c}{}_{\mathrm{i}}\text{(q)c}{}_{\mathrm{n?1}}\text{(q)q}{}^{\mathrm{i(n?i?1)}}$

is defined. It is conjectured that all coefficients in the polynomial C_n(q) are at most $\text{O(}\text{4}{}^{\mathrm{n}}\text{n}{}^3\text{)}$. It is shown that if this conjecture is true, then

$\text{S}{}_{\mathrm{n}}\text{<}\text{C}{}_3\text{4}{}^{\mathrm{n}}\text{n}{}^{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}$