Related papers: Neumaier graphs from cyclotomy with small coherent…

We construct a family of maximal linklessly embeddable graphs on $n$ vertices and $3n-5$ edges for all $n\ge 10$, and another family on $n$ vertices and $m< \frac{25n}{12}-\frac{1}{4}$ edges for all $n\ge 13$. The latter significantly…

This is the second report of our work on the construction of directed strongly regular graphs. In our previous work, we constructed a couple of infinite families of new directed strongly regular graphs on the sets of antiflags of partial…

We construct infinitely many signed graphs having symmetric spectrum, by using the NEPS and rooted product of signed graphs. We also present a method for constructing large cospectral signed graphs. Although the obtained family contains…

We prove that there exist infinite families of regular bipartite Ramanujan graphs of every degree bigger than 2. We do this by proving a variant of a conjecture of Bilu and Linial about the existence of good 2-lifts of every graph. We also…

We establish rigidity theorems for graph product von Neumann algebras $M_\Gamma=*_{v,\Gamma}M_v$ associated to finite simple graphs $\Gamma$ and families of tracial von Neumann algebras $(M_v)_{v\in\Gamma}$. We consider the following three…

We exhibit an infinite family of vector-valued mock theta functions indexed by positive integers coprime to $6$. These are built from specializations of Dyson's rank generating function and related functions studied by Watson, Gordon, and…

Let $q$ be an odd power of a prime $p$, and $S \subset \mathbb{F}_q^*$ such that $S=-S$ and $S/S \neq \mathbb{F}_q^*$. We show that the clique number of the Cayley graph $\operatorname{Cay}(\mathbb{F}_q^+,S)$ is at most…

It has long been known that there exist finite connected tetravalent arc-transitive graphs with arbitrarily large vertex-stabilisers. However, beside a well known family of exceptional graphs, related to the lexicographic product of a cycle…

For each $t \ge 1$ let $W_t$ denote the class of graphs other than stars that have diameter $2$ and contain neither a triangle nor a $K_{2,t}$. The famous Hoffman--Singleton Theorem implies that $W_2$ is finite. Recently Wood suggested the…

Euler graphs with only one (two) type(s) of cycles under (mod 4) operation were studied in Part-I(II). Here we consider the class of Euler graphs with only three types of cycles under (mod 4). This gives rise to four cases viz., graphs…

We show that the number of conjugacy classes of intersections $A\cap B^g$, for fixed finitely generated subgroups $A, B<F$ of a free group, is bounded above in terms of the ranks of $A$ and $B$; this confirms an intuition of Walter Neumann.…

Kriesel conjectured that every minimally $1$-tough graph has a vertex with degree precisely $2$. Katona and Varga (2018) proposed a generalized version of this conjecture which says that every minimally $t$-tough graph has a vertex with…

For every infinite sequence $\omega=x_1,x_2,...$, with $x_i\in\{0,1\}$, we construct an infinite 4-regular graph $X_{\omega}$. These graphs are precisely the Schreier graphs of the action of a certain self-similar group on the space…

We derive exact values and new bounds for the Shannon capacity of two families of graphs: the $q$-Kneser graphs and the tadpole graphs. We also construct a countably infinite family of connected graphs whose Shannon capacity is not attained…

In this paper, we study new Cayley graphs over the additive group of Galois rings. First we prove that they are expander graphs by using a Weil-Carlitz-Uchiyama type estimation of character sums for Galois rings. We also show that Cayley…

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 strongly regular graph is called trivial if it or its complement is a union of disjoint cliques. We prove that every infinite family of nontrivial strongly regular graphs is quasi-random in the sense of Chung, Graham and Wilson.

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…

Let G be a finitely presented group, and let {G_i} be a collection of finite index normal subgroups that is closed under intersections. Then, we prove that at least one of the following must hold: 1. G_i is an amalgamated free product or…

We describe an algorithm for the exhaustive generation of non-isomorphic graphs with a given number $k \ge 0$ of hamiltonian cycles, which is especially efficient for small $k$. Our main findings, combining applications of this algorithm…