图的离心率值列的研究（英文）

设G为一个图，对任意x∈V（G），其离心率e(x)定义为e(x)=max{d(x,u)│任意u∈（V（G）}。将G中各点的离心率的值按照（不重复）从小到大排列而得到的数列称为G的离心率值列。现设{ei}1 ≤i≤s为一个非减的整数数列。本得到了下面三个结果：（i){ei}1 ≤i≤s是图的离心率值列当且仅当{ei}1≤i≤s=e1,es]且e1≥1,es≤2e1;（ii）定义NG（e)={x│x∈V（G）且e(x)=e}，若│NG（e)│=1则e=r(G);(iii)有给定离心率值列r,r s]的图的最小阶fr,r s]为fr,r s]={2r s,若0≤s≤r-2;r s 1,若s=r-1或r；这里,s,s k]表示r,r s]数列{r-1 i}1≤i≤s 1。

On The Eccentricity Value Sequence of A Simple Graph

Let G be a simple graph. For x ∈ V(G) , the eccentricity e(x) of x is defined to be e(x) =max{d(x,u) u ∈ V(G) } . The sequence gotten by arranging the values of eccentricity in G in increasing order iscalled the eccentricity value sequence of G. Let {ei}1≤s be a nondecreasing integer sequence. The following results onare obtained in this paper.(I) {e,}1≤I≤s is a graphic eccentricity value sequence iff {ei}1≤I≤s = e1,es] and e1 ≥ 1, es ≤ 2e1 .(ii) Define NG(e) = {x1x ∈ V(G),e(x) = e} ,then I NG(e) = 1 only ifr(G) = e.(iii) The minimum order fr,r + s] of graphs with given eccentricity value sequence r,r + s] is fr,r+s]={2r+s,if0=s=r-2;r+s+1,if s=r-1or r; where r,r + s] = {r 1 + I}1≤I≤s+1 1.