中文摘要：

设G是一个连通图,最大度和最小度分别为△（G）和δ（G）.图G的非正则指标t（G）是指G的度序列中不同值的个数.如果t（G）=△（G）一δ（G）＋1,则称图G为极大非正则图.本文给出了极大非正则图和不含三角的极大非正则图边数的上界,同时给出极大非正则图边数的一个紧的下界.

英文摘要：

Let G be a connected graph with maximum degree △（G） and minimum degreeδ （G）. The irregularity index t（G） of G is defined as the number of distinct terms in the degree sequence of G. We say that G is maximally irregular if t（G） =△（G） - δ（G） ＋ 1. The purpose of this note is to establish upper bound on the size of maximally irregular graphs and triangle-free graphs and give a tight lower bound on the size of maximally irregular graphs.