Transitive,Locally Finite Median Graphs with Finite Blocks

The subject of this paper are infinite, locally finite, vertex-transitive median graphs. It is shown that the finiteness of the Θ-classes of such graphs does not guarantee finite blocks. Blocks become finite if, in addition, no finite sequence of Θ-contractions produces new cut-vertices. It is proved that there are only finitely many vertex-transitive median graphs of given finite degree with finite blocks. An infinite family of vertex-transitive median graphs with finite intransitive blocks is also constructed and the list of vertex-transitive median graphs of degree four is presented. Sandi Klavžar: Supported by the Ministry of Science of Slovenia under the grant P1-0297. The author is also with the Faculty of Mathematics and Natural Sciences, University of Maribor, Slovenia and the Institute of Mathematics, Physics and Mechanics, Ljubljana.     

Extensions of the Cauchy-Goursat Integral Theorem

It is natural to conjecture that if a function f is continuous on the closed region determined by a rectifiable 1-cycle Γ and complex-differentiable on the open region then Γ_∫f=0. The main result is an extension of the classical Cauchy-Goursat Theorem: the equality conjectured holds (with no boundary condition on f^′) under the additional hypothesis that the winding numbers of Γ define an Lp function and f satisfies a matching Hölder continuity condition near the image of Γ. (In particular, continuity suffices if p=∞.) The proof uses approximations of a rectifiable path by piecewise linear paths.     

关于Boutroux—Cartan定理的推广

推广了著名的Boutroux—Cartan定理。设aμ（μ=1，2，…，n）为复平面上任意的n个点，H为任意的一个正数，则在平面上同时使得n∏μ=1 ｜z-aμ｜≤（H/e）^n和n∑μ=1 1/｜z-aμ｜≥nlog（en）/H成立的点z可被含于总数不超过n，半径总和不超过2H的一组圈内。     

On Extensions of Myers' Theorem

Let M be a compact Riemannian manifold, and let h be a smoothfunction on M. Let p^h(x) = inf||–₁(Ric_x(,)–2Hess(h_x(,)).Here Ric_x denotes the Ricci curvature at x and Hess(h) is theHessian of h. Then M has finite fundamental group if ^h–p^h<0. Here ^h =:+2L_h is the Bismut-Witten Laplacian. This leadsto a quick proof of recent results on extension of Myers' theoremto manifolds with mostly positive curvature. There is also asimilar result for noncompact manifolds.     

Long arithmetic progressions in critical sets

Given a density 0<σ?1, we show for all sufficiently large primes p that if S⊆Z/pZ has the least number of three-term arithmetic progressions among all sets with at least σp elements, then S contains an arithmetic progression of length at least log^1/4+o(1)p.     

Polynomial relations between matrices of graphs

We derive a correspondence between the eigenvalues of the adjacency matrix and the signless Laplacian matrix of a graph when is -biregular by using the relation . This motivates asking when it is possible to have for a polynomial, , and matrices associated to a graph . It turns out that, essentially, this can only happen if is either regular or biregular.     

Extensions of the Erdös-Ko-Rado Theorem

An antichain of subsets of 1,2,...,n has the Erdös-Ko-Rado property if |A_i|?n/2 and A_i∩A_j≠Ø(i=j). This paper contains a number of results concerning the distribution of sizes of sets in such a family, and also in families where the restriction |A_i|?n/2 is removed.     

The Sturm-Hurwitz Theorem and its Extensions

The following principle is well-known in Harmonic Analysis: If a real function has a spectral gap at the origin then it must have many sign changes. We obtain some sharp estimates showing that the set of positivity of such functions cannot be too small. We also extend the principle above to complex functions: If a complex function has a spectral gap at the origin then the variation of argument of this function must be large.     

Two Extensions to Finsler's Recurring Theorem

Finsler's theorem asserts the equivalence of (i) and (ii) for pairs of real quadratic forms f and g on R ⁿ : (i) f( ξ ) >0 for all ξ≠ 0 with g( ξ ) =0; (ii) f-λ g>0 for some λ∈ R. We prove two extensions: 1. We admit a vector-valued quadratic form g: R ⁿ → R ^k , for which we show that (i) implies that f-λ . . . g>0 on an ( n-k+1 ) -dimensional subspace Y R ⁿ for some λ∈ R ^k . 2. In the nonstrict version of Finsler's theorem for indefinite g we replace R ⁿ by a real vector space X . Accepted 22 February 1998     

Degree distribution in random planar graphs

We prove that for each k?0, the probability that a root vertex in a random planar graph has degree k tends to a computable constant dk, so that the expected number of vertices of degree k is asymptotically dkn, and moreover that k_∑dk=1. The proof uses the tools developed by Giménez and Noy in their solution to the problem of the asymptotic enumeration of planar graphs, and is based on a detailed analysis of the generating functions involved in counting planar graphs. However, in order to keep track of the degree of the root, new technical difficulties arise. We obtain explicit, although quite involved expressions, for the coefficients in the singular expansions of the generating functions of interest, which allow us to use transfer theorems in order to get an explicit expression for the probability generating function p(w)=k_∑dkwk. From this we can compute the dk to any degree of accuracy, and derive the asymptotic estimate dk∼c⋅k−1/2qk for large values of k, where q≈0.67 is a constant defined analytically.     

Further results on random cubic planar graphs

We provide precise asymptotic estimates for the number of several classes of labeled cubic planar graphs, and we analyze properties of such random graphs under the uniform distribution. This model was first analyzed by Bodirsky and coworkers. We revisit their work and obtain new results on the enumeration of cubic planar graphs and on random cubic planar graphs. In particular, we determine the exact probability of a random cubic planar graph being connected, and we show that the distribution of the number of triangles in random cubic planar graphs is asymptotically normal with linear expectation and variance. To the best of our knowledge, this is the first time one is able to determine the asymptotic distribution for the number of copies of a fixed graph containing a cycle in classes of random planar graphs arising from planar maps.     

On types of growth for graph-different permutations

We consider an infinite graph G whose vertex set is the set of natural numbers and adjacency depends solely on the difference between vertices. We study the largest cardinality of a set of permutations of [n] any pair of which differ somewhere in a pair of adjacent vertices of G and determine it completely in an interesting special case. We give estimates for other cases and compare the results in case of complementary graphs. We also explore the close relationship between our problem and the concept of Shannon capacity “within a given type.”     

G-分次环中一个定理的推广

本推广单分次摸情形下的G-分次环的一个定理．     

柏龙树(THE PERRON TREE)定理的推广与应用

将平面上的柏龙树(the Perron Tree)定理推广了k维欧氏空间,由此推广了Fefferman定理.     

Extensions of Edelstein's Theorem on Contractive Mappings

The aim of this article is to provide extensions of Edelstein's theorem for a class of contractive mappings, namely cyclic contractive mappings. We prove the existence and convergence of best proximity points of a cyclic contractive map. We also discuss continuity properties of cyclic contractive maps. Finally, we give a characterization of such maps in the setting of a Hilbert space.     

Properties of two-dimensional sets with small sumset

We give tight lower bounds on the cardinality of the sumset of two finite, nonempty subsets A,B⊆R² in terms of the minimum number h₁(A,B) of parallel lines covering each of A and B. We show that, if h₁(A,B)?s and |A|?|B|?2s²−3s+2, then