Primitive Strongly Regular Triangle-Free Graphs with \(\mu=3\) do not Exist

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/b1924-8658-8779-p Primitive Strongly Regular Triangle-Free Graphs with \(\mu=3\) do not Exist Belousova, V. I. , Lee, H. , Makhnev, A. A. Vladikavkaz Mathematical Journal 2026. Vol. 28. Issue 3. Abstract: If \(\Gamma\) is an antipodal distance-regular graph of diameter 4 with strongly regular graph \(\Gamma_{1,2}\), then \(\Gamma_{3,4}\) is a strongly regular graph without triangles. For a graph \(\Gamma\) with intersection array \(\{20,18,3,1;1,3,18,20\}\), the graph \(\Gamma_{3,4}\) has parameters (162,21,0,3). Graphs of small degrees have intersection arrays \(\{20,18,3,1;1,3,18,20\}\), \(\{25,24,2,1;1,2,24,25\}\) and \(\{32,27,6,1;1,6,27,32\}\), \(\{36,25,2,1;1,2,25,36\}\). A graph with array \(\{32,27,6,1;1,6,27,32\}\) does not exist. Any strongly regular triangle-free graph \(\Gamma\) with \(\mu=3\) has parameters (162,21,0,3). A strongly regular graph with parameters \((162,21,0,3)\) is defined if and only if its bipartite doubling (a distance-regular graph with intersection array \(\{21,20,9,3,1;1,3,9,20,21\}\)) is defined. It is proved in the paper that a strongly regular graph with parameters \((162,21,0,3)\) does not exist. As a consequence, a distance-regular graph with intersection array \(\{20,18,3,1;1,3,18,20\}\) and a bipartite distance-regular graph with intersection array \(\{36,35,47,6;1,9,30,36\}\) do not exist. Keywords: distance-regular graph, strongly regular graph, antipodal graph Language: Russian Download the full text Download JATS XML For citation: Belousova, V. I., Lee, H. and Makhnev, A. A. Primitive Strongly Regular Triangle-Free Graphs with \(\mu=3\) do not Exist, Vladikavkaz Math. J., 2026, vol. 28, no. 3, pp.44-50 (in Russian). DOI 10.46698/b1924-8658-8779-p + References 1. Brouwer, A. E., Cohen, A. M. and Neumaier, A. Distance-Regular Graphs, Berlin, Heidelberg, New York, Springer-Verlag, 1989, 495 p. 2. Dalfó, C., Fiol, M. and Koolen, J. The Spectral Excess Theorem for Graphs with Few Eigenvalues Whose Distance-2 or Distance-1-or-2 Graph is Strongly Regular, Linear and Multilinear Algebra, 2018, vol. 67, no. 12, pp. 2373-2381. DOI: 10.1080/03081087.2018.1491944. 3. Makhnev, A. and Paduchikh, D. Inverse Problems in Distance-Regular Graphs of Diameter 4, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 1, pp. 199-208 (in Russian). DOI: 10.21538/0134-4889-2022-28-1-199-208. 4. Makhnev, A. and Paduchikh, D. Small AT4-Graphs and Strongly Regular Subgraphs Corresponding to Them, Proceedings of the Steklov Institute of Mathematics, 2017, vol. 296, no. 1, pp. 164-174. DOI: 10.1134/S0081543817020158. 5. Soicher, L. The Uniqueness of a Distance-Regular Graph with Intersection Array {32,27,8,1;1,4,27,32} and Related Results, Designs, Codes and Cryptography, 2017, vol. 84, no. 1-2, pp. 101-108. DOI: 10.1007/s10623-016-0223-6. 6. Coolsaet, K. and Jurišić, A. Using Equality in the Krein Conditions to Prove Nonexistence of Certain Distance-Regular Graphs, Journal of Combinatorial Theory, Series A, 2008, vol. 115, no. 6, pp. 1086-1095. DOI: 10.1016/j.jcta.2007.12.001. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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