Related papers: The strongly regular twisted $D_{5,5}(q)$ graph
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…
We generalize the notion of quasirandom which concerns a class of equivalent properties that random graphs satisfy. We show that the convergence of a graph sequence under the spectral distance is equivalent to the convergence using the…
Recent advances in Quantum Topology assign $q$-series to knots in at least three different ways. The $q$-series are given by generalized Nahm sums (i.e., special $q$-hypergeometric sums) and have unknown modular and asymptotic properties.…
Distance-regular graphs have many beautiful combinatorial properties. Distance-transitive graphs have very strong symmetries, and they are distance-regular, i.e. distance-transitivity implies distance-regularity. In this paper, we give…
Motivated by work of Haythorpe, Thomassen and the author showed that there exists a positive constant $c$ such that there is an infinite family of 4-regular 4-connected graphs, each containing exactly $c$ hamiltonian cycles. We complement…
We classify all the $2$-arc-transitive strongly regular graphs, and use this classification to study the family of finite $(G,3)$-geodesic-transitive graphs of girth $4$ or $5$ for some group $G$ of automorphisms. For this application we…
Characterizations graphs of some classes to induce periodic Grover walks have been studied for recent years. In particular, for the strongly regular graphs, it has been known that there are only three kinds of such graphs. Here, we focus on…
The family of bent functions is a known class of Boolean functions, which have a great importance in cryptography. The Cayley graph defined on $\mathbb{Z}_{2}^{n}$ by the support of a bent function is a strongly regular graph…
A set of graphs are called cospectral if their adjacency matrices have the same characteristic polynomial. In this paper we introduce a simple method for constructing infinite families of cospectral regular graphs. The construction is valid…
The first known families of cages arised from the incidence graphs of generalized polygons of order $q$, $q$ a prime power. In particular, $(q+1,6)$--cages have been obtained from the projective planes of order $q$. Morever, infinite…
The \emph{critical} group of a finite connected graph is an abelian group defined by the Smith normal form of its Laplacian. Let $q$ be a power of a prime and $H$ be a multiplicative subgroup of $K=\mathbb{F}_{q}$. By $\mathrm{Cay}(K,H)$ we…
A graph is called a strong (resp. weak) bar 1-visibility graph if its vertices can be represented as horizontal segments (bars) in the plane so that its edges are all (resp. a subset of) the pairs of vertices whose bars have a…
Let $G$ denote a $Q$-polynomial distance-regular graph with diameter $D$ at least 4. Assume that the intersection numbers of $G$ satisfy $a_i=0$ for $0 \leq i \leq D-1$ and $a_D\neq 0$. We show that $G$ is a polygon, a folded cube, or an…
We generalise recent results about quasi-Cartan, Cartan and diagonal subalgebras by introducing graded versions. We show that there is a correspondence between graded algebraic quasi-Cartan/ Cartan/ diagonal pairs and certain graded twisted…
We show that any connected regular graph with $d+1$ distinct eigenvalues and odd-girth $2d+1$ is distance-regular, and in particular that it is a generalized odd graph.
We introduce a new model of indeterminacy in graphs: instead of specifying all the edges of the graph, the input contains all triples of vertices that form a connected subgraph. In general, different (labelled) graphs may have the same set…
We consider here 6-regular plane graphs whose faces have size 1, 2 or 3. In Section 2 a practical enumeration method is given that allowed us to enumerate them up to 53 vertices. Subsequently, in Section 3 we enumerate all possible symmetry…
We find closed form formulas for Kemeny's constant and its relationship with two Kirchhoffian indices for some composite graphs that use as basic building block a graph endowed with one of several symmetry properties.
This paper completes the classification of triply-transitive strongly regular graphs, a program recently initiated by Herman, Maleki, and Razafimahatratra. By proving that the collinearity graph of the polar space $\mathcal{Q}^{-}(5,q)$ and…
A graph $G$ with $d+1$ distinct eigenvalues is called strongly distance-regular if $G$ itself is distance-regular, and its distance-$d$ graph $G_d$ is strongly-regular. In this note we provide a spectral characterization of those…