Related papers: There is no (75,32,10,16) strongly regular graph
Let $Q^+(2n+1,q)$ be a hyperbolic quadric of $\PG(2n+1,q)$. Fix a generator $\Pi$ of the quadric. Define $\cG_n$ as the graph with vertex set the points of $Q^+(2n+1,q)\setminus \Pi$ and two vertices adjacent if they either span a secant to…
A graph $G$ is a link-irregular graph if every two distinct vertices of $G$ have non-isomorphic links. The link of a vertex $v$ in $G$ is the subgraph induced by the neighbors of $v$ in $G$. Ali, Chartrand and Zhang [Discussiones…
In 2018 the first, Rukavina and the third author constructed with the aid of a computer the first example of a strongly regular graph $\Gamma$ with parameters (216, 40, 4, 8) and proved that it is the unique PSU(4,2)-invariant…
Edge-girth-regular graphs (abbreviated as \emph{egr} graphs) are regular graphs in which every edge is contained in the same number of shortest cycles. We prove that there is no $3$-regular \emph{egr} graph with girth $7$ such that every…
A strong clique in a graph is a clique intersecting every maximal independent set. We study the computational complexity of six algorithmic decision problems related to strong cliques in graphs and almost completely determine their…
Highly regular graphs for which not all regularities are explainable by symmetries are fascinating creatures. Some of them like, e.g., the line graph of W.~Kantor's non-classical $\mathrm{GQ}(5^2,5)$, are stumbling stones for existing…
Let $\mathcal{G}(4,2)$ be the set of connected regular graphs with four distinct eigenvalues in which exactly two eigenvalues are simple, $\mathcal{G}(4,2,-1)$ (resp. $\mathcal{G}(4,2,0)$) the set of graphs belonging to $\mathcal{G}(4,2)$…
We study combinatorial configurations with the associated point and line graphs being strongly regular. Examples not belonging to known classes such as partial geometries and their generalizations or elliptic semiplanes are constructed.…
In this paper we construct all strongly regular graphs, with at most 600 vertices, admitting a transitive action of the orthogonal group $O^+(6,2)$ or $O^-(6,2)$. Consequently, we prove the existence of strongly regular graphs with…
A graph is strongly perfect if every induced subgraph H has a stable set that meets every nonempty maximal clique of H. The characterization of strongly perfect graphs by a set of forbidden induced subgraphs is not known. Here we provide…
A Neumaier graph is a non-complete edge-regular graph containing a regular clique. In this paper we give some sufficient and necessary conditions for a Neumaier graph to be strongly regular. Further we show that there does not exist…
We determine that there is no partial geometry ${\cal G}$ with parameters $(s,t,\alpha)=(4,27,2)$. The existence of such a geometry has been a challenging open problem of interest to researchers for almost 40 years. The particular interest…
If $\Gamma$ is a graph for which every edge is in exactly one clique of order $\omega$, then one can form a new graph with vertex set equal to these cliques. This is a generalization of the line graph of $\Gamma$. We discover many general…
In the course of proving the strong perfect graph theorem, Chudnovsky, Robertson, Seymour, and Thomas showed that every perfect graph either belongs to one of five basic classes or admits one of several decompositions. Four of the basic…
We give simple arithmetic conditions that force the Sylow $p$-subgroup of the critical group of a strongly regular graph to take a specific form. These conditions depend only on the parameters $(v, k, \lambda, \mu)$ of the strongly regular…
We show that every 3-regular circle graph has at least two pairs of twin vertices; consequently no such graph is prime with respect to the split decomposition. We also deduce that up to isomorphism, K_4 and K_{3,3} are the only 3-connected,…
A $d$-regular graph on $n$ nodes has at most $T_{\max} = \frac{n}{3} \tbinom{d}{2}$ triangles. We compute the leading asymptotics of the probability that a large random $d$-regular graph has at least $c \cdot T_{\max}$ triangles, and…
We consider orbit partitions of groups of automorphisms for the symplectic graph and apply Godsil-McKay switching. As a result, we find four families of strongly regular graphs with the same parameters as the symplectic graphs, including…
A typical problem in optimal design theory is finding an experimental design that is optimal with respect to some criteria in a class of designs. The most popular criteria include the A- and D-criteria. Regular graph designs occur in many…
A regular clique in a regular graph is a clique such that every vertex outside of the clique is adjacent to the same positive number of vertices inside the clique. We continue the study of regular cliques in edge-regular graphs initiated by…