Classifying cubic symmetric graphs of order 10p or 10p~2

A graph is called s-regular if its automorphism group acts regularly on the set of its s-arcs. In this paper, the s-regular cyclic or elementary abelian coverings of the Petersen graph for each s ≥ 1 are classified when the fibre-preserving automorphism groups act arc-transitively. As an application of these results, all s-regular cubic graphs of order 10p or 10p2 are also classified for each s ≥ 1 and each prime p, of which the proof depends on the classification of finite simple groups.     

Classifying cubic symmetric graphs of order 10p or 10p 2

A graph is called s-regular if its automorphism group acts regularly on the set of its s-arcs. In this paper, the s-regular cyclic or elementary abelian coverings of the Petersen graph for each s ? 1 are classified when the fibre-preserving automorphism groups act arc-transitively. As an application of these results, all s-regular cubic graphs of order 10p or 10p ² are also classified for each s ? 1 and each prime p, of which the proof depends on the classification of finite simple groups.     

Regular automorphisms of order p2

Let G be a group, and let α be a regular automorphism of order p² of G, where p is a prime. If G is polycyclic-by-finite and the map ϕ : G →G defined by g^ϕ= [g,α] is surjective, then G is soluble. If G is polycyclic, then C_G(α^p) and G/[G,α^p] are both nilpotent-by-finite.     

Cubic semisymmetric graphs of order 8p~3

A regular edge-transitive graph is said to be semisymmetric if it is not vertex-transitive.By Folkman [J.Combin.Theory 3(1967),215-232],there is no semisymmetric graph of order 2p or 2p 2 for a prime p,and by Malni et al.[Discrete Math.274(2004),187-198],there exists a unique cubic semisymmetric graph of order 2p 3,the so called Gray graph of order 54.In this paper,it is shown that there is no connected cubic semisymmetric graph of order 4p 3 and that there exists a unique cubic semisymmetric graph of orde...     

Relative K homology and C* algebras

Let A be a separable nuclear C ⁺ algebra with unit. Let be a closed two-sided ideal in A. A relative K homology group K ⁰(A,) is defined. Closely related are topological definitions of properly supported K homology and of compactly supported relative K homology. Applications are to indices of Toeplitz operators and existence of coercive boundary conditions for elliptic differential operators.     

D(β)-vertex-distinguishing total coloring of graphs

A new concept of the D(β)-vertex-distinguishing total coloring of graphs, i.e., the proper total coloring such that any two vertices whose distance is not larger than β have different color sets, where the color set of a vertex is the set composed of all colors of the vertex and the edges incident to it, is proposed in this paper. The D(2)-vertex-distinguishing total colorings of some special graphs are discussed, meanwhile, a conjecture and an open problem are presented.     

The decomposition of Kv into K2×K5''''s

The decomposition of the complete graph Kv into Kr×Kc's, the products of Kr and Kc,is originated from the use of DNA library screening. In this paper, we consider the case where r=2 and c = 5, and show that such a decomposition exists if and only if v ≡ 1 (mod 25).     

Results on <Emphasis Type="Italic">F</Emphasis>-continuous graphs

For any nontrivial connected graph F and any graph G, the F-degree of a vertex v in G is the number of copies of F in G containing v. G is called F-continuous if and only if the F-degrees of any two adjacent vertices in G differ by at most 1; G is F-regular if the F-degrees of all vertices in G are the same. This paper classifies all P ₄-continuous graphs with girth greater than 3. We show that for any nontrivial connected graph F other than the star K _1,k , k ⩾ 1, there exists a regular graph that is not F-continuous. If F is 2-connected, then there exists a regular F-continuous graph that is not F-regular.      

On semisymmetric cubic graphs of order 6p~2

A graph r is said to be G-semisymmetric if it is regular and there exists a subgroup G of A := Aut(Г) acting transitively on its edge set but not on its vertex set. In the case of G. = A, we call r a semisymmetric graph. The aim of this paper is to investigate (G-)semisymmetric graphs of prime degree. We give a group-theoretical construction of such graphs, and give a classification of semisymmetric cubic graphs of order 6p2 for an odd prime p.     

A note on cyclic codes over GR(p2, m) of length pk

The number of self-dual cyclic codes of length p ^k over GR(p ², m) is determined by the nullity of a certain matrix M(p ^k , i ₁). With the aid of Genocchi numbers, we determine the nullity of M(p ^k , i ₁) and hence determine completely the number of such codes.     

Edge-transitive dihedral or cyclic covers of cubic symmetric graphs of order 2P

A regular cover of a connected graph is called dihedral or cyclic if its transformation group is dihedral or cyclic, respectively. Let X be a connected cubic symmetric graph of order 2p for a prime p. Several publications have investigated the classification of edge-transitive dihedral or cyclic covers of X for specific p. The edge-transitive dihedral covers of X have been classified for p=2 and the edge-transitive cyclic covers of X have been classified for p≤5. In this paper an extension of the above results to an arbitrary prime p is presented.     

A combinatorial proof of Euler-Fermat’s theorem on the representation of the primes p=8k+3 by the quadratic form x2+2y2

An elementary and extremely short proof of the theorem on the representation of the primes p = 8k + 3 by the quadratic form x₂ + 2y₂ with integers x and y. Bibliography: 1 title. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 330, 2006, pp. 155–157.     

Regularity of C1 smooth surfaces with prescribed p-mean curvature in the Heisenberg group

We consider a C ¹ smooth surface with prescribed p (or H)-mean curvature in the 3-dimensional Heisenberg group. Assuming only the prescribed p-mean curvature we show that any characteristic curve is C ² smooth and its (line) curvature equals  − H in the nonsingular domain. By introducing characteristic coordinates and invoking the jump formulas along characteristic curves, we can prove that the Legendrian (or horizontal) normal gains one more derivative. Therefore the seed curves are C ² smooth. We also obtain the uniqueness of characteristic and seed curves passing through a common point under some mild conditions, respectively. These results can be applied to more general situations.     

The newton filtration and d-determination of bifurcation problems related to C0 contact equivalence

In this paper, from the Newton filtration’s point of view, we construct the singular Riemannian metric and use the method in singular theory to study the bifurcation problems, and give the sufficient condition of d-determination of bifurcation problems with respect to C ⁰ contact equivalence. The special cases of the main result in this paper are the results of Sun Weizhi and Zou Jiancheng.     

The Atiyah-Segal completion theorem for C*-algebras

We generalize the Atiyah-Segal completion theorem to C ^*-algebras as follows. Let A be a C ^*-algebra with a continuous action of the compact Lie group G. If K _* ^G (A) is finitely generated as an R(G)-module, or under other suitable restrictions, then the I(G)-adic completion K _* ^G (A) is isomorphic to RK _*([A C(EG)]^G), where RK _* is representable K-theory for - C ^*-algebras and EG is a classifying space for G. As a corollary, we show that if and are homotopic actions of G, and if K _*(C ^* (G,A,)) and K _*(C ^* (G,A,)) are finitely generated, then K _*(C ^*(G,A,))K_*(C ^* (G,A,)). We give examples to show that this isomorphism fails without the completions. However, we prove that this isomorphism does hold without the completions if the homotopy is required to be norm continuous.This work was partially supported by an NSF Graduate Fellowship and by an NSF Postdoctoral Fellowship.     

Amply regular graphs and block designs

We study the amply regular diameter d graphs Γ such that for some vertex a the set of vertices at distance d from a is the set of points of a 2-design whose set of blocks consists of the intersections of the neighborhoods of points with the set of vertices at distance d-1 from a. We prove that the subgraph induced by the set of points is a clique, a coclique, or a strongly regular diameter 2 graph. For diameter 3 graphs we establish that this construction is a 2-design for each vertex a if and only if the graph is distance-regular and for each vertex a the subgraph Γ₃(a) is a clique, a coclique, or a strongly regular graph. We obtain the list of admissible parameters for designs and diameter 3 graphs under the assumption that the subgraph induced by the set of points is a Seidel graph. We show that some of the parameters found cannot correspond to distance-regular graphs.     

K-theory for rickart C*-algebras

We give an explicit index map for any properly infinite closed ideal of a Rickart C ^*-algebra. This generalizes Olsen's work on von Neumann algebras. We use our results to compute the topological and the algebraic K ₁-groups of any quotient algebra of a Rickart C ^*-algebra.     

Characterizing defect n-extendable graphs and -critical graphs

A near perfect matching is a matching saturating all but one vertex in a graph. Let G be a connected graph. If any n independent edges in G are contained in a near perfect matching where n is a positive integer and n(|V(G)|-2)/2, then G is said to be defect n-extendable. If deleting any k vertices in G where k|V(G)|-2, the remaining graph has a perfect matching, then G is a k-critical graph. This paper first shows that the connectivity of defect n-extendable graphs can be any integer. Then the characterizations of defect n-extendable graphs and (2k+1)-critical graphs using M-alternating paths are presented.     

Quasi-permutation Representations of the Groups SU (3, q2) and PSU (3, q2)

A square matrix over the complex field with non-negative integral trace is called a quasi-permutation matrix. For a finite group G the minimal degree of a faithful permutation representation of G is denoted by p(G). The minimal degree of a faithful representation of G by quasi-permutation matrices over the rational and the complex numbers are denoted by q(G) and c(G) respectively. Finally r(G) denotes the minimal degree of a faithful rational valued complex character of G. In this paper p(G), q(G), c(G) and r(G) are calculated for the groups PSU (3, q²) and SU (3, q²).AMS Subject Classification (2000): 20C15