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Table of Content

      
    20 February 2015
    Volume 50 Issue 02
    Disjoint 4-cycles and 8-cycles in graphs
    ZHANG Shao-hua, YAN Jin, LI Shuo
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2015, 50(02):  1-4.  doi:10.6040/j.issn.1671-9352.0.2014.282
    Abstract ( 408 )   PDF (1044KB) ( 443 )   Save
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    Let G be a graph of order 4k(k≥4) and δ (G)≥2k. Then G contains k-2 cycles of length 4 and a cycle of length 8 such that these k-1 cycles are disjoint. As an application, we prove that if G is a graph of order 4k and δ(G)≥2k, then at least one of the following two results is true: (1) G contains k-3 cycles of length 4 and a cycle of length 12; (2) G contains k-4 cycles of length 4 and two cycles of length 8, where the cycles are disjoint.
    Several conclusions of adjacent vertex distinguishing E-total coloring of the cartesian product graphs
    LIU Xin-sheng, DENG Wei-dong, WANG Zhi-qiang
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2015, 50(02):  5-8.  doi:10.6040/j.issn.1671-9352.0.2014.137
    Abstract ( 641 )   PDF (639KB) ( 780 )   Save
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    By using of the analysis method, the adjacent vertex distinguishing E-total coloring of the cartesian product graphs are studied, and the adjacent vertex distinguishing E-total chromatic numbers for the cartesian products of the graphs with chromatic number at least 2 or the graphs with adjacent vertex distinguishing E-total chromatic numbers 3 or 4 are discussed, some relevant conclusions are also obtained.
    Neighbor sum distinguishing total coloring of graphs with maximum degree 3 or 4
    YAO Jing-jing, XU Chang-qing
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2015, 50(02):  9-13.  doi:10.6040/j.issn.1671-9352.0.2014.362
    Abstract ( 526 )   PDF (650KB) ( 463 )   Save
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    A proper [k]-total coloring of a graph G is a map φ:VE→{1,2,…,k} such that φ(x)≠φ(y) for each pair of adjacent or incident elements x,yVE. Let f(v) denote the sum of the color of vertex v and the colors of the edges incident with v. A [k]-neighbor sum distinguishing total coloring of G is a [k]-total coloring of G such that for each edge uvE(G), f(u)≠f(v). Let tndiΣ(G) denote the smallest value k in such a coloring of G. Pil?niak and Wo?niak first introduced this coloring and conjectured that tndiΣ(G)≤Δ(G)+3 for any simple graph with maximum degree Δ(G). The maximum average degree of G is the maximum of the average degree of its non-empty subgraphs, which is denoted by mad(G). By using the Combinatorial Nullstellensatz and the discharging method, it is proved that if G is a graph with Δ(G)=3 and mad(G)<125, or Δ(G)=4 and mad(G)<52, then tndiΣ(G)≤Δ(G)+2.
    The algorithm for adjacent-vertex-distinguishing total coloring of graphs
    LI Jing-wen, JIA Xi-bei, DONG Wei, LI Xiao-hui, YAN Guang-hui
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2015, 50(02):  14-21.  doi:10.6040/j.issn.1671-9352.0.2014.145
    Abstract ( 503 )   PDF (768KB) ( 608 )   Save
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    With a proper total coloring of graph G, for any vertex v, its color set is made up of colors of v ertex vand all its incident edges. An adjacent vertex distinguishing total coloring of a graph G is a proper total coloring, such that any pair of adjacent vertices are incident to distinct sets of colors.The minimum coloring number is called the adjacent-vertex-distinguishing total chromatic number of G. According to adjacent-vertex-distinguishing total coloring rules, this paper presents a heuristic algorithm for the adjacent-vertex-distinguishing total coloring. The algorithm ascertains four sub-functions and one generic function and then iterates gradually in proper sequence with the help of the color matrix and complementary set. When the generic function value equals to zero, we say that the current coloring is successful. The experimental results show that the algorithm can obtain the chromatic number of the adjacent-vertex-distinguishing total coloring of graphs and the time complexity is not more than O(n3).
    General edge-coloring of mPn which is vertex distinguished by multisets
    GUO Hong-yuan, CHEN Xiang-en, WANG Zhi-wen
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2015, 50(02):  22-26.  doi:10.6040/j.issn.1671-9352.0.2014.334
    Abstract ( 343 )   PDF (928KB) ( 336 )   Save
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    Let G be a simple graph. A general edge-coloring of a graph G is an assignment of a number of colors to the edges. It is not necessary to assign two distinct colors to two adjacent edges. A general edge-coloring f of a graph G is called vertex distinguished by multisets, if, for any two distinct vertices u,v of a graph G, the multiset of the colors used to color the edges incident with u is different from the multiset of the colors used to color the edges incident with v. The minimum number of colors required for a general edge-coloring of G which is vertex distinguishing by multisets, denoted by c(G), is called the vertex distinguishing general edge chromatic number of G by multisets. Suppose mPn denotes the vertex-disjoint union of m paths of length n. The vertex distinguishing general edge-coloring(by multisets) of mPn will be discussed.
    Revised rainbow vertex-connection number of graphs
    WANG Wan-yu
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2015, 50(02):  27-31.  doi:10.6040/j.issn.1671-9352.0.2014.146
    Abstract ( 310 )   PDF (556KB) ( 329 )   Save
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    If for every pair u, v of distinct vertices, G contains a revised rainbow u-v geodesic, then G is revised strong vertex-connected. The minimum number k for which there exists a k-vertex-coloring of G that results in a revised strong rainbow vertex-connected graph is called the revised strong rainbow vertex-connection number of G, denoted by srvc*(G). Then rvc*(Cn)= for n≥4 is proved, and a upper bound is given for the revised strong rainbow connection number of graphs depending on the the number of edge-disjoint triangles.
    Positive and negative inertia indexes and nullity of two special kinds of tricyclic graphs
    YANG Chen, MA Hai-cheng
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2015, 50(02):  32-37.  doi:10.6040/j.issn.1671-9352.0.2014.267
    Abstract ( 622 )   PDF (610KB) ( 364 )   Save
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    The problem how to calculate the positive and negative inertia indexes and nullity of Ⅰ-type and Ⅱ-type tricyclic graphs are discussed. By deleting pendant trees and compressing internal paths, a method of calculating the positive and negative inertia indexes and nullity of the two kinds of tricyclic graphs is given. It is proved that the positive and negative inertia indexes and nullity of Ⅰ-type tricyclic graphs equal to the sum of some trees and bicyclic graphs(or unicyclic graphs or trees), respectively; the positive and negative inertia indexes and nullity of Ⅱ-type tricyclic graphs equal to the sum of some trees and simple tricyclic graphs, and the positive and negative inertia indexes and nullity of simple tricyclic graphs can be calculated by Matlab.
    On the vertex-arboricity of embedded graphs
    SUN Lin, CAI Hua
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2015, 50(02):  38-42.  doi:10.6040/j.issn.1671-9352.0.2014.324
    Abstract ( 442 )   PDF (963KB) ( 285 )   Save
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    An induced forest k-partition of a graph G is a k-partition (V1,V2,…,Vk) of the vertex set V(G) such that, for each i with 1≤i≤k, the induced subgraph G[Vi] is a forest. The vertex arboricity of a graph G is the minimum positive integer k such that G has an induced forest k-partition, denoted by va(G). Let G be a simple graph embedded in a surface of nonnegative Euler characteristic, and if G satisfies three kinds of conditions, then va(G)≤2.
    On the s-θ-completions for subgroups
    GAO Hui, GAO Sheng-zhe, YIN Li
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2015, 50(02):  43-46.  doi:10.6040/j.issn.1671-9352.0.2014.147
    Abstract ( 578 )   PDF (513KB) ( 354 )   Save
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    The notion of s-θ-completions of a proper subgroup of a finite group is introduced. Some new characterizations of the solvable, supersolvable groups are obtained by s-θ-completions of 2-maximal subgroups.
    The projective objects in the category of fuzzy quantales
    LU Jing, ZHAO Bin
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2015, 50(02):  47-54.  doi:10.6040/j.issn.1671-9352.0.2014.207
    Abstract ( 710 )   PDF (560KB) ( 272 )   Save
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    The concept of fuzzy weakly ⊗-stable completely distributive lattices is introduced. It is proved that the family of all down sets of a fuzzy ordered semigroup with a appropriate operation ⊗ is a fuzzy weakly ⊗-stable completely distributive lattice. A necessary and sufficient condition for a fuzzy completely distributive lattice to be a fuzzy weakly ⊗-stable completely distributive lattice is given. Finally, projective objects in the category of fuzzy quantales are studied. It is also proved that the E-projective objects in the category of fuzzy quantales are exactly the fuzzy weakly ⊗-stable completely distributive lattices.
    Eigenvalue asymptotics of a class of the Dirac operator
    HAO Ping-ping, WEI Guang-sheng
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2015, 50(02):  55-59.  doi:10.6040/j.issn.1671-9352.0.2014.268
    Abstract ( 558 )   PDF (519KB) ( 317 )   Save
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    In this paper, we are concered with the asymptotics of the eigenvalues of the classical Dirac opertor with the piecewise smooth potentials, and give the asymptotics of the eigenvalues up to n-2 term.
    General solution and stability of bi-cubic functional equation
    QI Wei-qing, JI Pei-sheng, LU Hai-ning
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2015, 50(02):  60-66.  doi:10.6040/j.issn.1671-9352.0.2014.349
    Abstract ( 563 )   PDF (551KB) ( 414 )   Save
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    Let X and Y be real vector spaces. A mapping f:X2Y is called bi-cubic if it satisfies f(2x1+x2,2y1+y2)+f(2x1+x2,2y1-y2)+f(2x1-x2,2y1+y2)+ f(2x1-x2,2y1-y2)=4f(x1+x2,y1+y2)+4f(x1-x2,y1+y2)+24f(x1,y1+y2)+ 4f(x1+x2,y1-y2)+4f(x1-x2,y1-y2)+24f(x1,y1-y2)+24f(x1+x2,y1)+ 24f(x1-x2,y1)+144f(x1,y1) for all x1,x2,y1,y2∈X. The solution of this equation is obtained and the Hyers-Ulam stability of it is proved on fuzzy Banach spaces.
    Wavelets method for solving system of fractional differential equations and the convergence analysis
    CHEN Yi-ming, KE Xiao-hong, HAN Xiao-ning, SUN Yan-nan, LIU Li-qing
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2015, 50(02):  67-74.  doi:10.6040/j.issn.1671-9352.0.2014.078
    Abstract ( 485 )   PDF (1236KB) ( 673 )   Save
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    The Legendre wavelets defined by the shifted Legendre polynomial is used to solve the numerical solution of the system of fractional differential equations with variable coefficient.The convergence analysis is presented to show that this method is correct for solving the fractional differential equations. Finally, three numerical examples are given to demonstrate the feasibility and efficiency of this method.
    Wajsberg's form of MTL algebras with applications
    KOU Hai-yan, WU Hong-bo
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2015, 50(02):  75-82.  doi:10.6040/j.issn.1671-9352.0.2014.342
    Abstract ( 553 )   PDF (545KB) ( 305 )   Save
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    MTL algebra is an important basic logic algebra. Firstly, the classical algebras forms of NMTL algebra is given by taking Wajsberg's method and some parts of axioms of MTL logic system, and it is proved that NMTL algebra and MTL algebra have identical structure. Secondly, it is proved that an NMTL algebras L satisfying the condition: ∀x,yL, x→y=(y→0)→(x→0) is BR0 algebra. Finally, it is proved that IMTL algebra and BR0 algebra have identical structure and the Wajsberg forms of BR0 algebra and BL algebra are given.
    Falling fuzzy ideals of BL-algebras
    YANG Yong-wei, HE Peng-fei, LI Yi-jun
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2015, 50(02):  83-89.  doi:10.6040/j.issn.1671-9352.0.2014.270
    Abstract ( 555 )   PDF (997KB) ( 484 )   Save
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    The paper establishes a theoretical approach to fuzzy (implicative) ideals of BL-algebras based on the theory of falling shadows. Then the relationships between falling fuzzy (implicative) ideals and fuzzy (implicative) ideals are investigated. It is proved that every fuzzy (implicative) ideal is a falling fuzzy (implicative) ideal, but not vice versa. Finally, some characterizations of falling fuzzy (implicative) ideals are obtained.
    The minimal and maximal ranks problems of matrix expression A-BXC under generalized symmetric constraints
    DAI Li-fang, LIANG Mao-lin, HE Wan-sheng
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2015, 50(02):  90-94.  doi:10.6040/j.issn.1671-9352.0.2014.227
    Abstract ( 463 )   PDF (521KB) ( 564 )   Save
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    Let R,S be generalized reflection matrices, that is, R*=R, R2=I, S*=S, S2=I. A matrix X is called generalized reflective matrix (generalized skew-reflective matrix) if RXS=X(RXS=-X). The minimal and maximal ranks problems of matrix expression A-BXC with generalized reflective or skew-reflective matrices X are studied, and some solvability conditions of matrix equation BXC=A are derived.