Abstract
A graph Γ is called a Deza graph if it is regular and the number of common neighbors of any two distinct vertices is one of two fixed values. A Deza graph is called a strictly Deza graph if it has diameter 2 and is not strongly regular. In 1992, Gardiner et al. proved that a strongly regular graph that contains a vertex with disconnected second neighborhood is a complete multipartite graph with parts of the same size greater than 2. In this paper, we study strictly Deza graphs with disconnected second neighborhoods of vertices. In Section 2, we prove that, if each vertex of a strictly Deza graph has disconnected second neighborhood, then the graph is either edge-regular or coedge-regular. In Sections 3 and 4, we consider strictly Deza graphs that contain at least one vertex with disconnected second neighborhood. In Section 3, we show that, if such a graph is edge-regular, then it is the s-coclique extension of a strongly regular graph with parameters (n, k, λ, μ), where s is an integer, s ≥ 2, and λ = μ. In Section 4, we show that, if such a graph is coedge-regular, then it is the 2-clique extension of a complete multipartite graph with parts of the same size greater than or equal to 3.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. V. Mityanina, “On K 1, 3-free strictly Deza graphs,” Proc. Steklov Inst. Math. 297 (Suppl. 1), S159–S162 (2017).
N. Biggs, Algebraic Graph Theory (Cambridge Univ. Press, Cambridge, 1993).
A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-Regular Graphs (Springer, Berlin, 1989).
A. E. Brouwer and J. H. van Lint, “Strongly regular graphs and partial geometries, enumeration and design,” in Proceedings of the Silver Jubilee Conference at the University of Waterloo, Ed. by D. M. Jackson and S. A. Vanstone (Academic, Toronto, 1984), pp. 85–122.
S. M. Cioaba and J. H. Koolen, “On the connectedness of the complement of a ball in distance-regular graphs,” J. Algebraic Combin. 38 (1), 191–195 (2013).
M. Erickson, S. Fernando, W. H. Haemers, D. Hardy, and J. Hemmeter, “Deza graphs: A generalization of strongly regular graphs,” J. Comb. Des. 7 (6), 395–405 (1999).
R. Diestel, Graph Theory (Springer, Berlin, 2010).
A. D. Gardiner, C. D. Godsil, A. D. Hensel, and G. F. Royle, “Second neighbourhoods of strongly regular graphs,” Discrete Math. 103, 161–170 (1992).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © S.V. Goryainov, G.S. Isakova, V.V. Kabanov, N.V.Maslova, L.V. Shalaginov, 2016, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2016, Vol. 22, No. 3.
Rights and permissions
About this article
Cite this article
Goryainov, S.V., Isakova, G.S., Kabanov, V.V. et al. On Deza graphs with disconnected second neighborhood of a vertex. Proc. Steklov Inst. Math. 297 (Suppl 1), 97–107 (2017). https://doi.org/10.1134/S008154381705011X
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S008154381705011X