For every two vertices u and v in a graph G,a u-v geodesic is a shortest path between u and v.Let I（u,v）denote the set of all vertices lying on a u-v geodesic.For a vertex subset S,let I（S） denote the union of all I（u,v）for u,v∈S.The geodetic number g（G）of a graph G is the minimum cardinality of a set S with I（S）=V（G）.For a digraph D,there is analogous terminology for the geodetic number g（D）.The geodetic spectrum of a graph G,denoted by S（G）,is the set of geodetic numbers of all orientations of graph G.The lower geodetic number is g-（G）=minS（G）and the upper geodetic number is g+（G）=maxS（G）.The main purpose of this paper is to study the relations among g（G）,g-（G）and g+（G）for connected graphs G.In addition,a sufficient and necessary condition for the equality of g（G）and g（G×K2）is presented,which improves a result of Chartrand,Harary and Zhang.
Chang-hong LU~(1,2) 1 Department of Mathematics,East China Normal University,Shanghai 200062,China
The eigenvalues of graphs play an important role in the fields of quantum chemistry,physics, computer science, communication network, and information science. Particularly,they can be interpreted in some situations as the energy levels of an electron in a molecule or as the possible frequencies of the tone of a vibrating membrane.The diameter of a graph,the maximum distance between any two vertices of a graph, has great impact on the service quality of communication networks. So we were motivated to investigate the sharp lower bound of the least eigenvalue of graphs with given diameter. Let n, d be the set of graphs on n vertices with diameter d. For any graph G∈n,d,by considering the least eigenvalue of its connected spanning bipartite subgraph, we obtained the sharp lower bound of the least eigenvalue of graph G. Furthermore,an upper bound of Laplacian spectral radius of graph G was given.
f:V(G)→{-1,0,1}称为图G的负全控制函数,如果对任意点v∈V,均有f[v]≥1,其中f[v]=sum from v∈N(v)f(u).如果对每个点v∈V,不存在负全控制函数g:V(G)→{-1,0,1},g≠f,满足g(v)≤f(v),则称f是一个极小负全控制函数.图的上负全控制数Γ_t^-(G)=max{w(f)|f是G的极小负全控制函数},其中w(f)=sum from v∈V(G)f(v).本文研究正则图的上负全控制数,证明了,令G是一个n阶r-正则图.若r为奇数,则Γ_t^-(G)≤(r^2+1)/(r^2+2r-1)n.
一张标记的图是订的对(G， L ) 由图 G 和它的标记的 L 组成：V (G) 鈫 ?{ 1， 2，鈥？n } ，在此 n =| V (G)| 。在一张标记的图的一条增加的 nonconsecutive 路径(G， L ) 任何一个是一条路径(u 1， u 2，鈥？u k )(k 鈮 ? 2 ) 在 G 以便 L (u i )+ 为所有 i = 的 2 鈮 ? L (u i+1 ) 1， 2，鈥？k？1 或顺序 1 的一条路径。增加 nonconsecutive 路径在的全部的数字(G， L ) 被 d 表示(G， L ) 。如果标记的 L 生产最大的 d，标记的 L 是最佳的(G， L ) 。在这份报纸，在 Zverovich (2004 ) 的比那简单的一个方法被给获得路径的最佳的标记。象周期和星那样的另外的特殊的图的最佳的标记被获得。
