Extremal Peisert-type graphs without the strict-EKR property

Let $q$ be a prime power. We study extremal Peisert-type graphs of order $q^2$ without the strict-EKR property, that is, Peisert-type graphs of order $q^2$ without the strict-EKR property and with the minimum number of edges. First, we determine this minimum number of edges for each value of $q$. If $q$ is a square, we show the uniqueness of extremal graph and its isomorphism with certain affine polar graph. Using the isomorphism, we conclude that there is no Hilton-Milner type result for this extremal graph. We also prove the tightness of the weight-distribution bound for both non-principal eigenvalues of this graph. If $q$ is a cube but not a square, we show the uniqueness of extremal graph and determine the number and the structure of non-canonical cliques. Finally, we show such uniqueness result does not extend to all $q$

Improvement of tools for government regulation of tourist activity based on analysis of simplicial complexes

The main aim of this research is conditioned by the need to improve efficiency of tourism in the context of economic modernization, which can lay the groundwork for sustainable development of both regions and the country. This is to be achieved using government regulation and support tools for sustainable development of tourist activity. The purpose of the article is to improve the existing tools for government regulation of tourist activity in the Russian Federation. The proposed method to investigate the problem is to analyze simplicial complexes of q-connectivity, which has allowed the authors to consider the structure of tools for government regulation of tourist activity and to identify their features and limitations. Because of the research, control elements of the structure of tools for government regulation of tourist activity have been identified, including: regulations to ensure safety of tourists and holidaymakers at resorts, advertising tourism opportunities, transportation of tourist flows, coordination with environmental institutions, and recreational nature management. The research has proved inconsistency of the existing structure of tools for government regulation of tourist activity, and, consequently, the need for its improvement based on sustainable development principles.peer-reviewe

A general construction of strictly Neumaier graphs and related switching

In this paper we propose a construction of Neumaier graphs with nexus 1, which generalises two known constructions. We then discuss small strictly Neumaier graphs obtained from the general construction and give a geometric description for some of them. Finally, we apply a variation of the Godsil-McKay switching to the general construction

Equitable partitions of Latin-square graphs

Funding: R.A. Bailey and Peter J. Cameron are grateful to Shanghai Jiao Tong University for funding, from the National Science Foundation of China (11671258) and STCSM (17690740800), a research visit where part of this study was done. Alexander L. Gavrilyuk was supported by BK21plus Center for Math Research and Education at Pusan National University, Republic of Korea. Sergey V. Goryainov was supported by the National Science Foundation of China, STCSM (17690740800) and RFBR (17‐51‐560008).We study equitable partitions of Latin-square graphs and give a complete classification of those whose quotient matrix does not have an eigenvalue -3.PostprintPeer reviewe