k-桥图匹配最大根的极值  

Extremum of the matching maximum root of k-bridge graphs

作　　者：马海成[1] 攸晓杰 MA Haicheng;YOU Xiaojie(School of Mathematics and Statistics,Qinghai Minzu University,Xining 810007,Qinghai,China)

机构地区：[1]青海民族大学数学与统计学院,青海西宁810007

出　　处：《山东大学学报（理学版）》2024年第6期19-24,共6页Journal of Shandong University(Natural Science)

基　　金：国家自然科学基金资助项目(11561056);青海省自然科学基金资助项目(2022-ZJ-924)。

摘　　要：设G是有n个点的图,μ(G,x)表示图G的匹配多项式,M_(1)(G)表示多项式μ(G,x)的最大根,称为匹配最大根。把k条路P_(a_(1)+2),P_(a_(2)+2),…,P_(a_(k)+2)的左右2个端点分别黏结成2个点后得到的图称为k-桥图,记为θ_(k)(a_(1),a_(2),…,a_(k))。有n个点且每一条路上的点数几乎相等的k-桥图记为θ_(k)^(*)(n)。证明了:在n个点的k-桥图中匹配最大根取得最小的图是θ_(k)^(*)(n),最大的图是θ_(k)(k-20,1,1…,1,n-k);在n个点的任意k-桥图中匹配最大根取得最小的图是2-桥图(圈)C_(n),最大的图是(n-1)-桥图θ_(n-1)(0,1,1…,1)。Let G be a graph with n vertices,andμ(G,x)denote the matching polynomial of graph G,M_(1)(G)denote the maximum root of the polynomialμ(G,x),which is called the matching maximum root.By identifying the first vertices and the last vertices of k paths P_(a_(1)+2),P_(a_(2)+2),…,P_(a_(k)+2),respectively,the resulting graph is called the k-bridge graphs,denoted byθ_(k)(a_(1),a_(2),…,a_(k)).A kbridge graph with n vertices and nearly equal number of vertices on each paths is denoted asθ_(k)^(*)(n).The following conclusions are proved.In all k-bridge graphs with n vertices,the matching maximum root to get the smallest graph isθ_(k)^(*)(n),and the biggest graph isθk(k-20,1,1,…,1,n-k).In any k-bridge graphs with n vertices,the graph that the matching maximum root to get the smallest is 2-bridge graphs Cn(cycle),and the biggest one is(n-1)-bridge graphθ_(n-1)(0,1,1…,1).

关 键 词：匹配多项式 匹配最大根 k-桥图 

分 类 号：O157.5[理学—数学]