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Bounds on the Closeness Centrality of a Graph
Journal article   Open access   Peer reviewed

Bounds on the Closeness Centrality of a Graph

Thomas Britz, Xin Hu, Abdellah Islam and Hopein C. Tang
BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY, v 48(4), 135
01 Jul 2025
DOI: https://doi.org/10.1007/s40840-025-01921-6
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https://doi.org/10.1007/s40840-025-01921-6View
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Abstract

Science & Technology Mathematics Physical Sciences
We present new values and bounds on the (normalised) closeness centrality C(sic)(C) of connected graphs and on its product l(sic)C(sic)(C) with the mean distance l(sic) of these graphs. Our main result presents the fundamental bounds 1 <= lC(sic)(C) < 2. We prove that the lower bound is tight and that the upper bound is asymptotically tight. Combining the lower bound with known upper bounds on the mean distance, we find ten new lower bounds for the closeness centrality of graphs. We also present explicit expressions for C(sic)(C) and l(sic)C(sic)(C) for specific families of graphs. Elegantly and perhaps surprisingly, the asymptotic values of nC(sic)(C) for paths P-n and ladder graphs L-n are both equal to pi, and the asymptotic limits of lC(C) for these families of graphs are both equal to pi /3. We conjecture that the set of values dense in the interval [1, 2). l(sic) C-C for all connected graphs is dense in the interval [1, 2).

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Mathematics
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