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On a novel eccentricity-based invariant of a graph

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Abstract

In this paper, for the purpose of measuring the non-self-centrality extent of non-selfcentered graphs, a novel eccentricity-based invariant, named as non-self-centrality number (NSC number for short), of a graph G is defined as follows: \(N\left( G \right) = \sum\nolimits_{{v_i}{v_j} \in V\left( G \right)} {|{e_i} - {e_j}|} \) where the summation goes over all the unordered pairs of vertices in G and e i is the eccentricity of vertex v i in G, whereas the invariant will be called third Zagreb eccentricity index if the summation only goes over the adjacent vertex pairs of graph G. In this paper, we determine the lower and upper bounds on N(G) and characterize the corresponding graphs at which the lower and upper bounds are attained. Finally we propose some attractive research topics for this new invariant of graphs.

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Authors and Affiliations

  1. College of Science, Nanjing University of Aeronautics & Astronautics, Nanjing, 210016, P. R. China

    Ke Xiang Xu

  2. Department of Mathematics, Sungkyunkwan University, Suwon, 440-746, Republic of Korea

    Kinkar Ch. Das

  3. Department of Mathematics, Faculty of Science, Selçuk University, Campus, Konya, 42075, Turkey

    Ayse Dilek Maden

Authors
  1. Ke Xiang Xu
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  2. Kinkar Ch. Das
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  3. Ayse Dilek Maden
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Correspondence to Ke Xiang Xu.

Additional information

Supported by NSFC (Grant No. 11201227), China Postdoctoral Science Foundation (Grant Nos. 2013M530253, 2014T70512), Natural Science Foundation of Jiangsu Province (Grant No. BK20131357), National Research Foundation funded by the Korean government (Grant Nos. 2013R1A1A2009341), TUBITAK and Scientific Research Project Office (BAP) of Selçuk University

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Xu, K.X., Das, K.C. & Maden, A.D. On a novel eccentricity-based invariant of a graph. Acta. Math. Sin.-English Ser. 32, 1477–1493 (2016). https://doi.org/10.1007/s10114-016-5518-z

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  • DOI: https://doi.org/10.1007/s10114-016-5518-z

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