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.
机构:
S China Normal Univ, Sch Math Sci, Guangzhou 510631, Guangdong, Peoples R ChinaS China Normal Univ, Sch Math Sci, Guangzhou 510631, Guangdong, Peoples R China
Zhu, Yingxue
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You, Lihua
Yang, Jieshan
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S China Normal Univ, Sch Math Sci, Guangzhou 510631, Guangdong, Peoples R ChinaS China Normal Univ, Sch Math Sci, Guangzhou 510631, Guangdong, Peoples R China
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Univ Ljubljana, Dept Math, Ljubljana 1000, Slovenia
Fac Informat Studies, Novo Mesto 8000, SloveniaFree Univ Berlin, Inst Informat, D-14195 Berlin, Germany
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S China Normal Univ, Sch Math Sci, Guangzhou 510631, Guangdong, Peoples R ChinaS China Normal Univ, Sch Math Sci, Guangzhou 510631, Guangdong, Peoples R China
Yang, Jieshan
Zhu, Yingxue
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S China Normal Univ, Sch Math Sci, Guangzhou 510631, Guangdong, Peoples R ChinaS China Normal Univ, Sch Math Sci, Guangzhou 510631, Guangdong, Peoples R China
Zhu, Yingxue
You, Zhifu
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Guangdong Polytech Normal Univ, Dept Comp Sci, Guangzhou 510665, Guangdong, Peoples R ChinaS China Normal Univ, Sch Math Sci, Guangzhou 510631, Guangdong, Peoples R China
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Inst Teknol Bandung, Fac Math & Nat Sci, Dept Math, Combinatorial Math Res Grp, Jalan Ganesha 10, Bandung 40132, Indonesia
Univ Islam Negeri Sunan Gunung Djati, Fac Sci & Technol, Dept Math, Bandung 40614, IndonesiaInst Teknol Bandung, Fac Math & Nat Sci, Dept Math, Combinatorial Math Res Grp, Jalan Ganesha 10, Bandung 40132, Indonesia
Ramdani, Rismawati
Salman, A. N. M.
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Inst Teknol Bandung, Fac Math & Nat Sci, Dept Math, Combinatorial Math Res Grp, Jalan Ganesha 10, Bandung 40132, IndonesiaInst Teknol Bandung, Fac Math & Nat Sci, Dept Math, Combinatorial Math Res Grp, Jalan Ganesha 10, Bandung 40132, Indonesia
Salman, A. N. M.
Assiyatun, Hilda
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Inst Teknol Bandung, Fac Math & Nat Sci, Dept Math, Combinatorial Math Res Grp, Jalan Ganesha 10, Bandung 40132, IndonesiaInst Teknol Bandung, Fac Math & Nat Sci, Dept Math, Combinatorial Math Res Grp, Jalan Ganesha 10, Bandung 40132, Indonesia
机构:
S China Normal Univ, Sch Math Sci, Guangzhou 510631, Guangdong, Peoples R ChinaS China Normal Univ, Sch Math Sci, Guangzhou 510631, Guangdong, Peoples R China
Yang, Jieshan
You, Zhifu
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Guangdong Polytech Normal Univ, Detartment Comp Sci, Guangzhou 510665, Guangdong, Peoples R ChinaS China Normal Univ, Sch Math Sci, Guangzhou 510631, Guangdong, Peoples R China