arXiv subjects
Publications and source records attributed to Reimbay Reimbayev.
In this paper, we examine the structure of strongly regular graphs with parameters $\lambda = 1$ and $\mu = 2$. In particular, we provide a complete classification of induced subgraphs of order seven and determine their relative frequencies. These findings contribute to a finer understanding of the local structure of such graphs and may be useful in related combinatorial and algebraic investigations.
An $srg(19,6,1,2)$ is the graph with the smallest parameter set in the family of strongly regular graphs with parameters $\lambda=1$ and $\mu=2$ for which the respective graph doesn't exist. The proof of that fact is based on algebraic arguments, particularly, on the Integrality Test, the very usefull tool for studying strongly regular graphs. To our best knowledge, there have not been proofs of pure combinatorial nature. In this short paper, we have decided to fill in this gap.
Strongly regular graphs are highly symmetrical and can be described fully with just a few parameters, yet the existence of many of them is still under the question. In this paper, we continue the study of the famuly of strongly regular graphs with parameters $\lambda =1$ and $\mu =2$ and establish all of their possible Hamiltonian subgraphs of order seven. By doing so we establish the lower and upper bounds for number of 7-gons, or 7-cycles, in such graphs.
Strongly regular graphs are highly symmetrical and can be described fully with just a few parameters yet the existence of many of them is still under the question. Due to this uncertainty, it is of immense interest to study their structure, in particular to obtain all the possible subgraphs of lower order. In this paper we study the family of strongly regular graphs with parameters $\lambda =1$ and $\mu =2$ and establish all their subgraphs of order six.
For a locally linear graph $G$, which is a graph built out of triangles, it is possible to construct another graph $G^*$ that would consist of triangles of $G$ as vertices, while sharing (or not sharing) a common vertex between a pair of triangles would define a binary relation for edges of $G^*$. In this paper we show that the spectrum of $G^*$ is uniquely defined by $G$. We will also show some structural similarities of these graphs; in particular, that the number of quadrilaterals and pentagons in both graphs are the same; that $G^*$ does not contain $K_4-e$ and $K_{1,4}$; and that $G$ can be reconstructed from $G^*$.
The existence of $srg(99,14,1,2)$ has been a question of interest for several decades to the moment. In this paper we consider the structural properties in general for the family of strongly regular graphs with parameters $\lambda =1$ and $\mu =2$. In particular, we establish the lower bound for the number of hexagons and, by doing that, we show the connection between the existence of the aforementioned graph and the number of its hexagons.