As a generalization of the scrambling index and the exponent,m-competition index has been widely applied to stochastic matrices,food webs and memoryless communication systems in recent years. For a positive integer m,...As a generalization of the scrambling index and the exponent,m-competition index has been widely applied to stochastic matrices,food webs and memoryless communication systems in recent years. For a positive integer m,where 1 ≤ m ≤ n,the mcompetition index( generalized competition index) of a primitive digraph D of order n is the smallest positive integer k such that for every pair of vertices x and y,there exist m distinct vertices v_1,v_2,…,v_m such that there exist walks of length k from x to v_i and from y to v_i for 1 ≤ i ≤ m. By analyzing the structure of θ-graphs( theta graphs) and using enumeration investigation methods,the mcompetition indices of primitive θ-graphs are studied and an upper bound is provided. Moreover, some corresponding extremal θ-graphs are characterized.展开更多
Woodall 所引入的一个图 G 的联结数 bind(G)是|ΓG(×)|/|×|的最小值,其中×遍取V(G)的所有子集(×≠φ).ΓG(×)表示 G 中×的邻点集(ΓG(×)≠V(G)).本文得到了θ图θ_(r,s,t)及其线圈的联结数:(1)若 n...Woodall 所引入的一个图 G 的联结数 bind(G)是|ΓG(×)|/|×|的最小值,其中×遍取V(G)的所有子集(×≠φ).ΓG(×)表示 G 中×的邻点集(ΓG(×)≠V(G)).本文得到了θ图θ_(r,s,t)及其线圈的联结数:(1)若 n=r+s+t-1是偶数,则 bind(θ_(r,s,t)=1.(2)若 n 是奇数:当 r,s,t 均为偶数时,bind(θ_(r,s,t))=(r+s+t-2)/r+s+t;否则,bind(θ_(r,s,t,)=(r+s+t-2)/r+s+t-3.(3)若 r,s,t 中至少有一个大于2,则 bind(L(θ_(r,s,t,)=(r+s+t-1)/r+s+t-2.展开更多
基金Shanxi Scholarship Council of China(No.2012-070)Foundation of North University of China(No.2013-12-1)
文摘As a generalization of the scrambling index and the exponent,m-competition index has been widely applied to stochastic matrices,food webs and memoryless communication systems in recent years. For a positive integer m,where 1 ≤ m ≤ n,the mcompetition index( generalized competition index) of a primitive digraph D of order n is the smallest positive integer k such that for every pair of vertices x and y,there exist m distinct vertices v_1,v_2,…,v_m such that there exist walks of length k from x to v_i and from y to v_i for 1 ≤ i ≤ m. By analyzing the structure of θ-graphs( theta graphs) and using enumeration investigation methods,the mcompetition indices of primitive θ-graphs are studied and an upper bound is provided. Moreover, some corresponding extremal θ-graphs are characterized.
