Classifying cubic symmetric graphs of order 10p or 10p2

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.     

Connected cubic <Emphasis Type="Italic">s</Emphasis>-arc-regular Cayley graphs of finite nonabelian simple groups

A graph is said to be s-arc-regular if its full automorphism group acts regularly on the set of its s-arcs. In this paper, we investigate connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups. Two sufficient and necessary conditions for such graphs to be 1- or 2-arcregular are given and based on the conditions, several infinite families of 1- or 2-arc-regular cubic Cayley graphs of alternating groups are constructed. This work was supported by Guangxi Science Foundations (Grant No. 0832054) and Guangxi Postgraduate Education Innovation Research (Grant No. 2008105930701M102)     

Acyclic edge colorings of planar graphs and seriesparallel graphs

A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G, denoted by a′(G), is the least number of colors in an acyclic edge coloring of G. Alon et al. conjectured that a′(G) ⩽ Δ(G) + 2 for any graphs. For planar graphs G with girth g(G), we prove that a′(G) ⩽ max{2Δ(G) − 2, Δ(G) + 22} if g(G) ⩾ 3, a′(G) ⩽ Δ(G) + 2 if g(G) ⩾ 5, a′(G) ⩽ Δ(G) + 1 if g(G) ⩾ 7, and a′(G) = Δ(G) if g(G) ⩾ 16 and Δ(G) ⩾ 3. For series-parallel graphs G, we have a′(G) ⩽ Δ(G) + 1. This work was supported by National Natural Science Foundation of China (Grant No. 10871119) and Natural Science Foundation of Shandong Province (Grant No. Y2008A20).     

On the Classification of Arc-transitive Circulant Digraphs of Order Odd-Prime-Squared

A Cayley graph F = Cay（G, S） of a group G with respect to S is called a circulant digraph of order pk if G is a cyclic group of the same order. Investigated in this paper are the normality conditions for arc-transitive circulant （di）graphs of order p^2 and the classification of all such graphs. It is proved that any connected arc-transitive circulant digraph of order p^2 is, up to a graph isomorphism, either Kp2, G（p^2,r）, or G（p,r）[pK1], where r｜p- 1.     

A [k,k+1]-factor containing given Hamiltonian cycle

Letk⩾2 be an integer and let G be a graph of ordern with minimum degree at leastk, n⩾8k -16 for evenn and n⩾6k - 13 for oddn. If the degree sum of each pair of nonadjacent vertices of G is at least n, then for any given Hamiltonian cycleC. G has a [k, k + 1]-factor containingC Preject supported partially by an exchange program between the Chinese Academy of Sciences and the Japan Society for Promotion of Sciences and by the National Natural Science Foundation of China (Grant No. 19136012)     

Adjacent strong edge colorings and total colorings of regular graphs

It is conjectured that χas(G) = χt(G) for every k-regular graph G with no C5 component (k 2). This conjecture is shown to be true for many classes of graphs, including: graphs of type 1; 2-regular, 3-regular and (|V (G)| - 2)-regular graphs; bipartite graphs; balanced complete multipartite graphs; k-cubes; and joins of two matchings or cycles.     

Large deviations for heavy-tailed random sums of independent random variables with dominatedly varying tails

We prove large deviation results on the partial and random sums Sn = ∑i=1n Xi,n≥1; S(t) = ∑i=1N(t) Xi, t≥0, where {N(t);t≥0} are non-negative integer-valued random variables and {Xn;n≥1} are independent non-negative random variables with distribution, Fn, of Xn, independent of {N(t); t≥0}. Special attention is paid to the distribution of dominated variation.     

On Hua-Tuan’s conjecture

Let G be a finite group and |G| = pn, p be a prime. For 0 m n, sm(G) denotes the number of subgroups of of order pm of G. Loo-Keng Hua and Hsio-Fu Tuan have ever conjectured: for an arbitrary finite p-group G, if p > 2, then sm(G) ≡ 1, 1 + p, 1 + p + p2 or 1 + p + 2p2 (mod p3). In this paper, we investigate the conjecture, and give some p-groups in which the conjecture holds and some examples in which the conjecture does not hold.     

On the value distribution of <Emphasis Type="Italic">ƒ</Emphasis> + <Emphasis Type="Italic">a</Emphasis>(<Emphasis Type="Italic">ƒ</Emphasis>′)<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let ƒ be a transcendental meromorphic function, a a nonzero finite complex number, and n ⩾ 2 a positive integer. Then ƒ + a(ƒ′) ⁿ assumes every complex value infinitely often. This answers a question of Ye for n = 2. A related normality criterion is also given. This work was supported by the National Natural Science Foundation of China (Grant No. 10771076), the Natural Science Foundation of Guangdong Province, China (Grant No. 07006700) and by the German-Israeli Foundation for Scientific Research and Development (Grant No. G-809-234.6/2003)     

Semisymmetric cubic graphs as regular covers of K 3,3

A regular graph X is called semisymmetric if it is edge-transitive but not vertex-transitive. For G ≤ AutX, we call a G-cover X semisymmetric if X is semisymmetric, and call a G-cover X one-regular if Aut X acts regularly on its arc-set. In this paper, we give the sufficient and necessary conditions for the existence of one-regular or semisymmetric Zn-Covers of K3,3. Also, an infinite family of semisymmetric Zn×Zn-covers of K3,3 are constructed.     

Cubic vertex‐transitive graphs of order 2pq

A graph is vertex?transitive or symmetric if its automorphism group acts transitively on vertices or ordered adjacent pairs of vertices of the graph, respectively. Let G be a finite group and S a subset of G such that 1?S and S={s^?1 | s∈S}. The Cayleygraph Cay(G, S) on G with respect to S is defined as the graph with vertex set G and edge set {{g, sg} | g∈G, s∈S}. Feng and Kwak [J Combin Theory B 97 (2007), 627–646; J Austral Math Soc 81 (2006), 153–164] classified all cubic symmetric graphs of order 4p or 2p² and in this article we classify all cubic symmetric graphs of order 2pq, where p and q are distinct odd primes. Furthermore, a classification of all cubic vertex‐transitive non‐Cayley graphs of order 2pq, which were investigated extensively in the literature, is given. As a result, among others, a classification of cubic vertex‐transitive graphs of order 2pq can be deduced. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 285–302, 2010     

Tripling Construction for Large Sets of Resolvable Directed Triple Systems

In this paper, we first define a doubly transitive resolvable idempotent quasigroup （DTRIQ）, and show that aDTRIQ of order v exists if and only ifv ≡0（mod3） and v ≠ 2（mod4）. Then we use DTRIQ to present a tripling construction for large sets of resolvable directed triple systems, which improves an earlier version of tripling construction by Kang （J. Combin. Designs, 4 （1996）, 301-321）. As an application, we obtain an LRDTS（4·3^n） for any integer n ≥ 1, which provides an infinite family of even orders.     

On graphs with the largest Laplacian index

Let G be a connected simple graph on n vertices. The Laplacian index of G, namely, the greatest Laplacian eigenvalue of G, is well known to be bounded above by n. In this paper, we give structural characterizations for graphs G with the largest Laplacian index n. Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on n and k for the existence of a k-regular graph G of order n with the largest Laplacian index n. We prove that for a graph G of order n ⩾ 3 with the largest Laplacian index n, G is Hamiltonian if G is regular or its maximum vertex degree is Δ(G) = n/2. Moreover, we obtain some useful inequalities concerning the Laplacian index and the algebraic connectivity which produce miscellaneous related results. The first author is supported by NNSF of China (No. 10771080) and SRFDP of China (No. 20070574006). The work was done when Z. Chen was on sabbatical in China.     

On the order of certain close to regular graphs without a matching of given size

A graph G is a {d, d+k}-graph, if one vertex has degree d+k and the remaining vertices of G have degree d. In the special case of k = 0, the graph G is d-regular. Let k, p ⩾ 0 and d, n ⩾ 1 be integers such that n and p are of the same parity. If G is a connected {d, d+k{-graph of order n without a matching M of size 2|M| = n − p, then we show in this paper the following: If d = 2, then k ⩾ 2(p + 2) and

Cayley’s decomposition and pólya’sW-property of ordinary linear differential equations

A proof is given for the equivalence of Pólya’sW- property of a linear differential equationL _n (D) y=0 to the possibility of decomposingL _n (D) ≡ Π _n ¹ [D+λ_i(x)] in a given interval. In this case a set ofn independent solutions form a Chebyshev system in the interval. An application determines intervals of non-oscillation for solutions of linear equations of the second order. Research supported by the National Science Foundation Grant No. GP-3897 with the University of Maryland.     

Global dimension and left derived functors of Hom

It is well known that the right global dimension of a ring R is usually computed by the right derived functors of Hom and the left projective resolutions of right R-modules. In this paper, for a left coherent and right perfect ring R, we characterize the right global dimension of R, from another point of view, using the left derived functors of Hom and the right projective resolutions of right R-modules. It is shown that rD(R)≤n (n≥2) if and only if the gl right Proj-dim MR≤n - 2 if and only if Extn-1(N, M) = 0 for all right R-modules N and M if and only if every (n - 2)th Proj-cosyzygy of a right R-module has a projective envelope with the unique mapping property. It is also proved that rD(R)≤n (n≥1) if and only if every (n-1)th Proj-cosyzygy of a right R-module has an epic projective envelope if and only if every nth Vroj-cosyzygy of a right R-module is projective. As corollaries, the right hereditary rings and the rings R with rD(R)≤2 are characterized.     

A Boundedness Criterion via Atoms for Linear Operators in Hardy Spaces

Let p∈(0,1] and s≥[n(1/p−1)], where [n(1/p−1)] denotes the maximal integer no more than n(1/p−1). In this paper, the authors prove that a linear operator T extends to a bounded linear operator from the Hardy space H ^p (ℝ ⁿ ) to some quasi-Banach space ℬ if and only if T maps all (p,2,s)-atoms into uniformly bounded elements of ℬ.      

DEGREE CONDITIONS OF INDUCED MATCHING EXTENDABLE GRAPHS

Abstract. A simple graph G is induced matching extendable,shortly IM-extendable,if every in-duced matching of G is included in a perfect matching of G. The degree conditions of IM-extend-able graphs are researched in this paper. The main results are as follows：     

On cubic non‐Cayley vertex‐transitive graphs

In 1983, the second author [D. Maru?i?, Ars Combinatoria 16B (1983), 297–302] asked for which positive integers n there exists a non‐Cayley vertex‐transitive graph on n vertices. (The term non‐Cayley numbers has later been given to such integers.) Motivated by this problem, Feng [Discrete Math 248 (2002), 265–269] asked to determine the smallest valency ?(n) among valencies of non‐Cayley vertex‐transitive graphs of order n. As cycles are clearly Cayley graphs, ?(n)?3 for any non‐Cayley number n. In this paper a goal is set to determine those non‐Cayley numbers n for which ?(n) = 3, and among the latter to determine those for which the generalized Petersen graphs are the only non‐Cayley vertex‐transitive graphs of order n. It is known that for a prime p every vertex‐transitive graph of order p, p² or p³ is a Cayley graph, and that, with the exception of the Coxeter graph, every cubic non‐Cayley vertex‐transitive graph of order 2p, 4p or 2p² is a generalized Petersen graph. In this paper the next natural step is taken by proving that every cubic non‐Cayley vertex‐transitive graph of order 4p², p>7 a prime, is a generalized Petersen graph. In addition, cubic non‐Cayley vertex‐transitive graphs of order 2p^k, where p>7 is a prime and k?p, are characterized. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 77–95, 2012     

Equienergetic self-complementary graphs

In this paper equienergetic self-complementary graphs on p vertices for every p = 4k, k ⩾ 2 and p = 24t + 1, t ⩾ 3 are constructed.