On Crown-Free Graphs with Regular μ-Subgraphs,II

The crown is the complete multipartite graph K _1,1,3. Terwilliger graphs without crowns and graphs without 3-cocliques with regular -subgraphs of given positive degree are studied. As a corollary, the local structure of graphs in which the neighborhoods of vertices are regular Terwilliger graphs of diameter 2 and some of these neighborhoods contain no 7-paws is determined. Connected crown-free graphs in which -subgraphs are edge regular graphs of diameter not exceeding 2 with given parameters are described.     

On Regular, Strongly Regular Congruences on Ordered Semigroups

Ｓｉｍｉｌａｒｔｏｔｈｅｏｒｙｏｆｃｏｎｇｒｕｅｎｃｅｓｏｎｓｅｍｉｇｒｏｕｐｓ，ｃｏｎｇｒｕｅｎｃｅｓｏｎｏｒｄｅｒｅｄｓｅｍｉｇｒｏｕｐｓｐｌａｙａｎｉｍｐｏｒｔａｎｔｒｏｌｅｉｎｓｔｕｄｙｉｎｇｔｈｅｓｔｒｕｃｔｕｒｅｓｏｆｏｒｄｅｒｅｄｓｅｍ...     

Strongly Regular Rings and SF-rings

in this paper, new characteristic properties of strongly regular rings are' given.Relations between certain generalizations of duo rings are also considered. The followingconditions are shown to be equivalent: (1) R is a strongly regular ring; (2) R is a left SFring such that every product of two independent closed left ideals of R is zero; (3) R is aright SF-ring such that every product of two independent closed left ideals of R is zero; (4)R is a left SF-ring whose every special left annihilator is a quasi-ideal; (5) R is a right SFring whose every special left annihilator is a quasi-ideal; (6) R is a left SF-ring whose everymaximal left ideal is a quasi-ideal; (7) R is a right SF-ring whose every maximal left ideal isa quasi-ideal; (8) R is a left SF-ring such that the set N(R) of all nilpotent elements of R isa quasi-ideal; (9) R is a right SF-ring such that N(R) is a quasi-ideal.     

Properties of Regular Uniform k—Stratified Graphs

A regular graph G = (V, E) is a k-stratified graph if V is partitioned into V₁, V₂, …, V_k subsets called strata. The stratification splits the degree d_v ∀v ϵ V into k-integers d_v1, d_v2, …, d_vk each one corresponding to a stratum. If d_v1 = d_v2 = … = d_vk ∀v ϵ V then G is called regular uniform k-stratified, RU^k_s(n, d) where n is the cardinality of the vertex set in each stratum and d is the degree of every vertex in each stratum. For every k, the class RU^k_s(n, d) has a unique graph generator class RU^l_s(n, d) derived by decomposition of graphs in RU^k_s(n, d). We investigate the minimization of the cardinality of V, the colorability, vertex coloring and the diameter of the graphs in the class. We also deal with complexity questions concerning RU^k_s(n, d). Some well-known computer network models such as barrel shifters and hypercubes are shown to belong in RU^k_s(n, d).     

Classification of Regular Planar Graphs with Diameter Two

In the present paper, the regular planar graphs with diameter two are classified.     

Longest Cycles in 2-Connected Regular Claw-Free Graphs

All graph considered In this paper are undirected and finite,without loops ormultip1e edges.A graph is called claw-free if it does not contain a copy of K_(1,3)as an induced subgraph.For a graph C,let δ(G)denote the minimum degree of G.There have been a great many results in recent years dealing with longest cycles.for example, M.Matthews and D.Summer have shown the following result:     

On Graphs with Zero Determinant of Adjacency Matrices

Ｏｎ　Ｇｒａｐｈｓ　ｗｉｔｈ　Ｚｅｒｏ　Ｄｅｔｅｒｍｉｎａｎｔ　ｏｆ　Ａｄｊａｃｅｎｃｙ　ＭａｔｒｉｃｅｓＸｕＹｉｎｆｅｎｇ（徐寅峰）（ＴｈｅＳｃｈｏｏｌｏｆＭａｎａｇｅｍｅｎｔ，Ｘｉ＇ａｎＪｉａｏｔｏｎｇＵｎｉｖｅｒｓｉｔｙ，Ｘｉ＇ａｎ，７１００４...     

On Some Total Chromatic Number ofOuterplanar Graphs with △(G) =6

ＯｎＳｏｍｅＴｏｔａｌＣｈｒｏｍａｔｉｃＮｕｍｂｅｒｏｆＯｕｔｅｒｐｌａｎａｒＧｒａｐｈｓｗｉｔｈ△（Ｇ）＝６￥ＺｈａｎｇＺｈｏｎｇｆｕ；ＬｕＸｉｎｚｈｏｎｇ；ＨａｎＪｉｎｃａｎｇ；ＬｉｎＫｅｒｏｎｇ（ＬａｎｚｈｏｕＲａｉｌｗａｙＩｎｓｔｉｔｕｔｅ，...     

Orderly Algorithm to Enumerate Central Groupoids and Their Graphs

A graph has the unique path property UPPn if there is a unique path of length n between any ordered pair of nodes. This paper reiterates Royle and MacKay＇s technique for constructing orderly algorithms. We wish to use this technique to enumerate all UPP2 graphs of small orders 3^2 and 4^2. We attempt to use the direct graph formalism and find that the algorithm is inefficient. We introduce a generalised problem and derive algebraic and combinatoric structures with appropriate structure. Then we are able to design an orderly algorithm to determine all UPP2 graphs of order 3^2, which runs fast enough. We hope to be able to determine the UPP2 graphs of order 4^2 in the near future.     

On Edge Singularity and Eigenvectors of Mixed Graphs

Let G be a mixed glaph which is obtained from an undirected graph by orienting some of its edges. The eigenvalues and eigenvectors of G are, respectively, defined to be those of the Laplacian matrix L（G） of G. As L（G） is positive semidefinite, the singularity of L（G） is determined by its least eigenvalue λ1 （G）. This paper introduces a new parameter edge singularity εs（G） that reflects the singularity of L（G）, which is the minimum number of edges of G whose deletion yields that all the components of the resulting graph are singular. We give some inequalities between εs（G） and λ1 （G） （and other parameters） of G. In the case of εs（G） = 1, we obtain a property on the structure of the eigenvectors of G corresponding to λ1 （G）, which is similar to the property of Fiedler vectors of a simple graph given by Fiedler.     

On All Fractional （a,b, k）-Critical Graphs

Let a,b,k,r be nonnegative integers with 1≤a≤b and r≥2.LetG be a graph of order n with n(a+b)(r(a+b)-2)+ak/a.In this paper,we first show a characterization for all fractional(a,b,k)-critical graphs.Then using the result,we prove that G is all fractional(a,b,k)-critical if δ(G)≥(r-1)b2/a+k and |NG(x1)∪NG(x2)∪···∪NG(xr)|≥bn+ak/a+b for any independent subset {x1,x2,...,xr} in G.Furthermore,it is shown that the lower bound on the condition|NG(x1)∪NG(x2)∪···∪NG(xr)|≥bn+ak/a+b is best possible in some sense,and it is an extension of Lu's previous result.     

On Growth and Covering Theorem for Strongly Spirallike Mapping of Type β with Order α

In this paper, we deduce growth and covering theorem for f(x) by the other means,where f(x) is strongly spirallike mapping of typeβwith orderαdefined on Unit Ball B of complex Banach space. and still give growth upper bound and distortion upper bound for subordinate mapping.     

On codimensions k immersions of m-manifolds for k = 1 and k = m − 2

Let us consider M a closed smooth connected m-manifold, N a smooth (2m − 2)-manifold and a continuous map, with . We prove that if is injective, then f is homotopic to an immersion. Also we give conditions to a map between manifolds of codimension one to be homotopic to an immersion. This work complements some results of Biasi et al. (Manu. Math. 104, 97–110, 2001; Koschorke in The singularity method and immersions of m-manifolds into manifolds of dimensions 2m − 2, 2m − 3 and 2m − 4. Lecture Notes in Mathematics, vol. 1350. Springer, Heidelberg, 1988; Li and Li in Math. Proc. Camb. Phil. Soc. 112, 281–285, 1992).     

On Coflat Modules and Their Hereditary Properties

In[4]Hiremath introduced the notion of copureness in the category of R-modules.We define on this basis coflatness of modules and study some of theirhereditary properties dual to that of the flat modules.The later can be used tocharacterize a class of rings larger than that of semihereditary ones.     

On the k—smoothness and k—strong smoothness

5l.IntroductionFrompal,er[11,L2],to[3j,tI1efol1owingresultsareobtained:1)LetXbeaBanachspace,thenxes(X)={xeX;llxIl=1}isG-differetiableifandonIyif,foreachyeX,letXI=span(x,y).2)LetXbeaBanachspace,thenxeS(X)isF-differetiableifandonlyifIn[4j,NanchaoxunandWangjianhuaintroducedk-smoothnessandk-strongsmooth-ness.lnthisPaPer,wegive:Theorem1LetXbeaBanachspace,thcnthefollowingstatementsareequivalent.(i)xo6S(X)isk-smooth.(ii)IfX*cyXisk-dimensionalsubspaceandx,eX*,thenwhereU(x,,r)={xeX;l1x-xo…     

On Some Total Chromatic Number ofOuterplanar Graphs with Δ(G) = 6(English)

AMS(1991)SubjectClassifiction:05C15Definition1Aplanargraphissaidtobeanouterplanarifitsvenicescanbeplacedontheboundaryofaparticularface,calledtheouterface.Definition2Aplanaredge-face--colouringofaplanargraphG(V,E,F)isanassignmentofkcolourstoalledgesandfacesinEUFsuchthatnotwoadjacentorincidentelementsreceivethesamecolour.Theminimumnumberkforwhichaproperdege-facek-colouringofGexistsiscalledtheedge--facetotalchromaticnumberofG,denotedbyX.(G).similarly,wecandefinethevertex-edge--facetotalchr…     

The Signless Laplacian Spectral Radius of Tricyclic Graphs with k Pendant Vertices

In this paper,we determine the unique graph with the largest signless Laplacian spectral radius among all the tricyclic graphs with n vertices and k pendant vertices.