CAJ | 학술논문

图的无符号拉普拉斯矩阵是图的邻接矩阵和度对角矩阵的和,其特征值记为q1≥q2≥…≥qn.设C(n,m)是由n个顶点m条边的连通图构成的集合,这里1≤n-1≤m≤(n2).如果对于任意的G∈C(n,m)都有q1(G*)≥q1(G)成立,图G*∈C(n,m)叫做最大图.这篇文章证明了对任意给定的正整数a=m-n+1,如果n＞-1/2+a+1/2√(1+12a12a2),那么n＜ q1(G*)＜n+1,进而得到,对任意的G∈C(n,m),只要n＞-1/2+a+1/2√(1+12a12a2),就有q1(G)＜n+1.
도적무부호랍보랍사구진시도적린접구진화도대각구진적화,기특정치기위q1≥q2≥…≥qn.설C(n,m)시유n개정점m조변적련통도구성적집합,저리1≤n-1≤m≤(n2).여과대우임의적G∈C(n,m)도유q1(G*)≥q1(G)성립,도G*∈C(n,m)규주최대도.저편문장증명료대임의급정적정정수a=m-n+1,여과n＞-1/2+a+1/2√(1+12a12a2),나요n＜ q1(G*)＜n+1,진이득도,대임의적G∈C(n,m),지요n＞-1/2+a+1/2√(1+12a12a2),취유q1(G)＜n+1.
The signless Laplacian matrix of a graph is defined to be the sum of its adjacency matrix and degree diagonal matrix,and its eigenvalues are denoted by q1 ≥q2 ≥ … ≥ qn.Let C(n,m) be a set of connected graphs in which every graph has n vertices and m edges,where 1 ≤ n - 1 ≤ m ≤ (n2).A graph G* ∈ C(n,m) is called maximum if q1(G*) ≥ q1(G) for any G ∈ C(n,m).In this paper,we proved that for any given positive integer a=m-n+1,n ＜ q1(G*) ＜ n+1 if n＞-1/2+a+1/2√(1+12a+12a2),which leads to q1(G) ＜ n + 1 for any G ∈ C(n,m) whenever n ＞ -1/2 + a + 1/2√(1 + 12a + 12a2).