Graphs $\Gamma$ of diameter 4 for which $\Gamma_{3,4}$ is a strongly regular graph with $\mu=4,6$
Ural mathematical journal, Tome 10 (2024) no. 1, pp. 76-83
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We consider antipodal graphs $\Gamma$ of diameter 4 for which $\Gamma_{1,2}$ is a strongly regular graph. A.A. Makhnev and D.V. Paduchikh noticed that, in this case, $\Delta=\Gamma_{3,4}$ is a strongly regular graph without triangles. It is known that in the cases $\mu=\mu(\Delta)\in \{2,4,6\}$ there are infinite series of admissible parameters of strongly regular graphs with $k(\Delta)=\mu(r+1)+r^2$, where $r$ and $s=-(\mu+r)$ are nonprincipal eigenvalues of $\Delta$. This paper studies graphs with $\mu(\Delta)=4$ and 6. In these cases, $\Gamma$ has intersection arrays $\{{r^2+4r+3},{r^2+4r},4,1;1,4,r^2+4r,r^2+4r+3\}$ and $\{r^2+6r+5,r^2+6r,6,1;1,6,r^2+6r,r^2+6r+5\}$, respectively. It is proved that graphs with such intersection arrays do not exist.
Keywords:
Distance-regular graph, Strongly regular graph, Triple intersection numbers
@article{UMJ_2024_10_1_a6,
author = {Alexander A. Makhnev and Mikhail P. Golubyatnikov and Konstantin S. Efimov},
title = {Graphs $\Gamma$ of diameter 4 for which $\Gamma_{3,4}$ is a strongly regular graph with $\mu=4,6$},
journal = {Ural mathematical journal},
pages = {76--83},
publisher = {mathdoc},
volume = {10},
number = {1},
year = {2024},
language = {en},
url = {http://geodesic.mathdoc.fr/item/UMJ_2024_10_1_a6/}
}
TY - JOUR
AU - Alexander A. Makhnev
AU - Mikhail P. Golubyatnikov
AU - Konstantin S. Efimov
TI - Graphs $\Gamma$ of diameter 4 for which $\Gamma_{3,4}$ is a strongly regular graph with $\mu=4,6$
JO - Ural mathematical journal
PY - 2024
SP - 76
EP - 83
VL - 10
IS - 1
PB - mathdoc
UR - http://geodesic.mathdoc.fr/item/UMJ_2024_10_1_a6/
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%A Mikhail P. Golubyatnikov
%A Konstantin S. Efimov
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%J Ural mathematical journal
%D 2024
%P 76-83
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%N 1
%I mathdoc
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Alexander A. Makhnev; Mikhail P. Golubyatnikov; Konstantin S. Efimov. Graphs $\Gamma$ of diameter 4 for which $\Gamma_{3,4}$ is a strongly regular graph with $\mu=4,6$. Ural mathematical journal, Tome 10 (2024) no. 1, pp. 76-83. http://geodesic.mathdoc.fr/item/UMJ_2024_10_1_a6/