Abstract: The classification of graphs of diameter $2$ by the number of pairs of diametral vertices contained in the graph is designed. All possible values of the parameters $n$ and $k$ are established for which there exists a $n$-vertex graph of diameter $2$ that has exactly $k$ pairs of diametral vertices. As a corollary, the smallest order of these graphs is found. Such graphs with a large number of vertices are also described and counted. In addition, for any fixed integer $k\geq 1$ inside each distinguished class of $n$-vertex graphs of diameter $2$ containing exactly $k$ pairs of diametral vertices, a class of typical graphs is constructed. For the introduced classes, the almost all property is studied for any $k=k(n)$ with the growth restriction under consideration, covering the case of a fixed integer $k\geq 1$. As a consequence, it is proved that it is impossible to limit the number of pairs of diametral vertices by a given fixed integer $k$ in order to obtain almost all graphs of diameter $2$.

Cite: Fedoryaeva T.I.
Classification of graphs of diameter $2$
Сибирские электронные математические известия (Siberian Electronic Mathematical Reports). 2020. Т.17. С.502-512. DOI: 10.33048/semi.2020.17.031 WOS Scopus РИНЦ MathNet OpenAlex

Dates:

Submitted:	Jan 19, 2020
Published online:	Apr 6, 2020

Identifiers:

Web of science:	WOS:000525533100001
Scopus:	2-s2.0-85129198773
Elibrary:	44726543
MathNet:	rus/semr1226
OpenAlex:	W3020593852

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DB	Citing
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