Related papers: On a family of diamond-free strongly regular graph…

In this paper we consider the question of when a strongly regular graph with parameters $((s+1)(st+1),s(t+1),s-1,t+1)$ can exist. These parameters arise when the graph is derived from a generalized quadrangle, but there are other examples…

In this paper, we assume that $q>0$, $p>1$ and $s\in(0,1)$ , and consider the following nonlinear fractional p-Laplacian equations on finite graphs: \begin{equation*} \left\{ \begin{array}{lll} \partial_t u^q(x,t)+(-\Delta)_p^su=0,\\[15pt]…

We study pseudo-geometric strongly regular graphs whose second subconstituent with respect to a vertex is a cover of a strongly regular graph or a complete graph. By studying the structure of such graphs, we characterize all graphs…

This paper discusses the complete structure of the (bull, diamond)-free graphs. As an application of that, we give the characterization of the partitionable (bull, diamond)-free graphs. Moreover, we show that such a partition for a…

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…

The strong fractional choice number of a graph $G$ is the infimum of those real numbers $r$ such that $G$ is $(\lceil rm \rceil, m)$-choosable for every positive integer $m$. The strong fractional choice number of a family ${\cal G}$ of…

Let $H(n, q^2)$ be a non-degenerate Hermitian variety of $PG(n, q^2)$, $n \ge 2$. Let NU$(n+1, q^2)$ be the graph whose vertices are the points of $PG(n, q^2) \setminus H(n, q^2)$ and two vertices $P_1$, $P_2$ are adjacent if the line…

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…

We give a general construction leading to different non-isomorphic families $\Gamma_{n,q}(\K)$ of connected $q$-regular semisymmetric graphs of order $2q^{n+1}$ embedded in $\PG(n+1,q)$, for a prime power $q=p^h$, using the linear…

A graphical regular representation (GRR) of a group $G$ is a Cayley graph of $G$ whose full automorphism group is equal to the right regular permutation representation of $G$. Towards a proof of the conjecture that only finitely many finite…

A graphical regular representation (GRR) of a group $G$ is a Cayley graph of $G$ whose full automorphism group is equal to the right regular permutation representation of $G$. In this paper we study cubic GRRs of $\mathrm{PSL}_{n}(q)$…

The quadrance between two points A_1=(x_1, y_1) and A_2=(x_2, y_2) is the number Q (A_1, A_2) = (x_1 - x_2)^2 + (y_1 - y_2)^2. Let q be an odd prime power and F_q be the finite field with $q$ elements. The unit-quadrance graph D_q has the…

This is a report on a failed attempt to construct new graphs that are strongly regular with no triangles. The approach is based on the assumption that the second subconstituent has an equitable partition with four parts. For infinitely many…

In this paper, we study prime order automorphisms of generalized quadrangles. We show that, if $\mathcal{Q}$ is a thick generalized quadrangle of order $(s,t)$, where $s > t$ and $s+1$ is prime, and $\mathcal{Q}$ has an automorphism of…

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…

We consider a primitive distance-regular graph $\Gamma$ with diameter at least $3$. We use the intersection numbers of $\Gamma$ to find a positive semidefinite matrix $G$ with integer entries. We show that $G$ has determinant zero if and…

In a projective plane $\Pi_q$ of order $q$, a non-empty point set ${\cal S}_t$ is a $t$-semiarc if the number of tangent lines to ${\cal S}_t$ at each of its points is $t$. If ${\cal S}_t$ is a $t$-semiarc in $\Pi_q$, $t<q$, then each line…

Let $G$ be a bridgeless graph. In 2011 Kung and Royle showed that all roots of the flow polynomial $F(G,\lambda)$ of $G$ are integers if and only if $G$ is the dual of a chordal and plane graph. In this article, we study whether a…

If S is a set of q+2 points in P^2(F_q) such that some point of S is not on any line containing two other points of S, then in suitable coordinates S has the form S_f:={(c:f(c):1) : c in F_q} U {(1:0:0),(0:1:0)} for some f(X) in F_q[X]. Let…

Let $S$ be the collection of quadratic polynomial maps, and degree $2$-rational maps whose automorphism groups are isomorphic to $C_2$ defined over the rational field. Assuming standard conjectures of Poonen and Manes on the period length…