NONREGULAR GRAPHS WITH MINIMAL TOTAL IRREGULARITY

被引:8
|
作者
Abdo, Hosam [1 ]
Dimitrov, Darko [2 ]
机构
[1] Free Univ Berlin, Dept Math & Comp Sci, D-14195 Berlin, Germany
[2] Hsch Tech & Wirtschaft Berlin, Dept Engn Sci 1, D-12459 Berlin, Germany
关键词
total irregularity; extremal graphs;
D O I
10.1017/S0004972715000271
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
The total irregularity of a simple undirected graph G is defined as irrt(G) = 1/2 Sigma(u,v)is an element of V(G) vertical bar d(G)(u) - d(G)(v)vertical bar, where dG(u) denotes the degree of a vertex u is an element of V(G). Obviously, irr(t)(G) = 0 if and only if G is regular. Here, we characterise the nonregular graphs with minimal total irregularity and thereby resolve the recent conjecture by Zhu et al. ['The minimal total irregularity of graphs', Preprint, 2014, arXiv: 1404.0931v1] about the lower bound on the minimal total irregularity of nonregular connected graphs. We show that the conjectured lower bound of 2n-4 is attained only if nonregular connected graphs of even order are considered, while the sharp lower bound of n-1 is attained by graphs of odd order. We also characterise the nonregular graphs with the second and the third smallest total irregularity.
引用
收藏
页码:1 / 10
页数:10
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