# Article

 Title: Radius-invariant graphs (English) Author: Bálint, V. Author: Vacek, O. Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 (print) ISSN: 2464-7136 (online) Volume: 129 Issue: 4 Year: 2004 Pages: 361-377 Summary lang: English . Category: math . Summary: The eccentricity $e(v)$ of a vertex $v$ is defined as the distance to a farthest vertex from $v$. The radius of a graph $G$ is defined as a $r(G)=\min _{u \in V(G)}\lbrace e(u)\rbrace$. A graph $G$ is radius-edge-invariant if $r(G-e)=r(G)$ for every $e \in E(G)$, radius-vertex-invariant if $r(G-v)= r(G)$ for every $v \in V(G)$ and radius-adding-invariant if $r(G+e)=r(G)$ for every $e \in E(\overline{G})$. Such classes of graphs are studied in this paper. (English) Keyword: radius of graph Keyword: radius-invariant graphs MSC: 05C12 MSC: 05C35 MSC: 05C75 idZBL: Zbl 1080.05505 idMR: MR2102610 DOI: 10.21136/MB.2004.134047 . Date available: 2009-09-24T22:16:21Z Last updated: 2020-07-29 Stable URL: http://hdl.handle.net/10338.dmlcz/134047 . Reference: [1] Buckley, F., Harary, F.: Distance in Graphs.Addison-Wesley, Redwood City, 1990. Reference: [2] Buckley, F., Itagi K. M., Walikar, H. B.: Radius-edge-invariant and diameter-edge-invariant graphs.Discrete Math. 272 (2003), 119–126. MR 2019205, 10.1016/S0012-365X(03)00189-4 Reference: [3] Buckley, F., Lewinter, M.: Graphs with all diametral paths through distant central nodes.Math. Comput. Modelling 17 (1990), 35–41. MR 1236507, 10.1016/0895-7177(93)90250-3 Reference: [4] Dutton, R. D., Medidi, S. R., Brigham, R. C.: Changing and unchanging of the radius of graph.Linear Algebra Appl. 217 (1995), 67–82. MR 1322543 Reference: [5] Frucht, R., Harary, F.: On the corona of two graphs.Aequationes Math. 4 (1970), 322–325. MR 0281659, 10.1007/BF01844162 Reference: [6] Gliviak, F.: On radially extremal graphs and digraphs, a survey.Math. Bohem. 125 (2000), 215–225. Zbl 0963.05072, MR 1768809 Reference: [7] Graham, N., Harary, F.: Changing and unchanging the diameter of a hypercube.Discrete Appl. Math. 37/38 (1992), 265–274. MR 1176857, 10.1016/0166-218X(92)90137-Y Reference: [8] Harary, F.: Changing and unchanging invariants for graphs.Bull. Malaysian Math. Soc. 5 (1982), 73–78. Zbl 0512.05035, MR 0700121 Reference: [9] Vizing, V. G.: The number of edges in a graph of given radius.Dokl. Akad. Nauk 173 (1967), 1245–1246. (Russian) Zbl 0158.42504, MR 0210622 .

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