Related papers: Strongly regular graphs with parameters (81,30,9,1…

While standard Weisfeiler-Leman vertex labels are not able to distinguish even vertices of regular graphs, there is proposed and tested family of inexpensive polynomial time vertex and edge invariants, distinguishing much more difficult…

This paper studies bounds in a strong form of regularity for $3$-uniform hypergraphs which was developed by Frankl, Gowers, Kohayakawa, Nagle, R\"{o}dl, Skokan, and Schacht. Regular decompositions of this type involve two structural…

In this paper, we show that if G is strongly regular then the Gallai graph and the anti-Gallai graph of G are edge-regular. We also identify conditions under which the Gallai and anti-Gallai graphs are themselves strongly regular, as well…

In 2010, Koolen and Bang proposed the following conjecture: For a fixed integer $m \geq 2$, any geometric distance-regular graph with smallest eigenvalue $-m$, diameter $D \geq 3$ and $c_2 \geq 2$ is either a Johnson graph, a Grassmann…

Three new graph invariants are introduced which may be measured from a quantum graph state and form examples of a framework under which other graph invariants can be constructed. Each invariant is based on distinguishing a different number…

Let $Q^{+}(2n-1,2)$ be a non-degenerate hyperbolic quadric of $PG(2n-1,2)$. Let $NO^{+}(2n,2)$ be the tangent graph, whose vertices are the points of $PG(2n-1,2) \setminus Q^{+}(2n-1,2)$ and two vertices $u,~v$ are adjacent if the line…

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…

This thesis focuses on theoretical and algorithmic tools for determining the numbers of induced subgraphs in strongly regular graphs, SRGs, and on further applications of such numbers. We consider in more detail a restricted class of these…

A pair of graphs $(\Gamma,\Sigma)$ is called unstable if their direct product $\Gamma\times\Sigma$ admits automorphisms not from $\mathrm{Aut}(\Gamma)\times\mathrm{Aut}(\Sigma)$, and such automorphisms are said to be unexpected. The…

We study characteristics which might distinguish two-graphs by introducing different numerical measures on the collection of graphs on $n$ vertices. Two conjectures are stated, one using these numerical measures and the other using the deck…

In [Distrance-regular Cayley graphs on dihedral groups, J. Combin. Theory Ser B 97 (2007) 14--33], Miklavi\v{c} and Poto\v{c}nik proposed the problem of characterizing distance-regular Cayley graphs, which can be viewed as an extension of…

Steinerberger defined a notion of boundary for a graph and established a corresponding isoperimetric inquality. Hence, "large" graphs have more boundary vertices. In this paper, we first characterize graphs with three boundary vertices in…

For all Artin groups, we characterise the girth (i.e. the length of a shortest cycle) of the defining graph algebraically, showing that it is an isomorphism invariant. Using this result, we prove that the Artin groups based on a cycle graph…

The paper provides the first constructions of strongly regular graphs and association schemes from weakly regular plateaued functions over finite fields of odd characteristic. We generalize the construction method of strongly regular graphs…

The structural parameter twin-width was introduced by Bonnet et al. in [FOCS 2020], and already this first paper included an asymptotic argument bounding the twin-width of planar graphs by a non-explicit constant. Quite recently, we have…

We consider signed graphs, i.e, graphs with positive or negative signs on their edges. We determine the admissible parameters for the $\{5,6,\ldots,10\}$-regular signed graphs which have only two distinct eigenvalues. For each obtained…

Distance-regular graphs are a class of regualr graphs with pretty combinatorial symmetry. In 2007, Miklavi\v{c} and Poto\v{c}nik proposed the problem of charaterizing distance-regular Cayley graphs, which can be viewed as a natural…

We report about the results of the application of modern computer algebra tools for construction of directed strongly regular graphs. The suggested techniques are based on the investigation of non-commutative association schemes and Cayley…

In this paper we show that there does not exist a strongly regular graph with parameters $(1911,270,105,27)$.

In this paper, we examine the structure of vertex- and edge-transitive strongly regular graphs, using normal quotient reduction. We show that the irreducible graphs in this family have quasiprimitive automorphism groups, and prove (using…