Related papers: Designs from Paley graphs and Peisert graphs

We study the spectrum of generalized Paley graphs $\Gamma(k,q)=Cay(\mathbb{F}_q,R_k)$, undirected or not, with $R_k=\{x^k:x\in \mathbb{F}_q^*\}$ where $q=p^m$ with $p$ prime and $k\mid q-1$. We first show that the eigenvalues of…

We consider generalized Paley graphs $\Gamma(k,q)$, generalized Paley sum graphs $\Gamma^+(k,q)$, and their corresponding complements $\bar \Gamma(k,q)$ and $\bar \Gamma^+(k,q)$, for $k=3,4$. Denote by $\Gamma = \Gamma^*(k,q)$ either…

A path in an edge-colored graph is called a proper path if no two adjacent edges of the path are colored the same. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to…

A straight-line drawing of a graph $G$ is a mapping which assigns to each vertex a point in the plane and to each edge a straight-line segment connecting the corresponding two points. The rectilinear crossing number of a graph $G$,…

For $q\in\mathbb{R}$, the $Q$-matrix $Q=Q_q$ of a connected simple graph $G=(V,E)$ is $Q_q=(q^{\partial(x,y)})_{x,y\in V}$, where $\partial$ denotes the path-length distance. Describing the set $\pi(G)$ consisting of those $q\in \mathbb{R}$…

The family of generalized Paley graphs of prime power order $q$ and degree $(q-1)/k$ is studied. It is shown that the automorphism group of a graph in this family is a subgroup of ${\mathrm{A\Gamma L}}(1,q)$ whenever $q$ is sufficiently…

Let $B$ and $R$ be two simple graphs with vertex set $V$, and let $G(B,R)$ be the simple graph with vertex set $V$, in which two vertices are adjacent if they are adjacent in at least one of $B$ and $R$. For $X \subseteq V$, we denote by…

This paper studies the strong fractional choice number $ch^s_f(G)$ and the strong fractional paint number $\chi^s_{f,P}(G)$ of a graph $G$. We prove that these parameters of any finite graph are rational numbers. On the other hand, for any…

Answering a question of Kolaitis and Kopparty, we show that, for given integer $q>1$ and pairwise nonisomorphic connected graphs $G_1...G_k$, if $p=p(n) $ is such that $\Pr(G_{n,p}\supseteq G_i)\to 1$ $\forall i$, then, with $\xi_i$ the…

Let $G_n$ be a random geometric graph, and then for $q,p \in [0,1)$ we construct a "$(q,p)$-perturbed noisy random geometric graph" $G_n^{q,p}$ where each existing edge in $G_n$ is removed with probability $q$, while and each non-existent…

Let $G$ be a graph without 4-cycles and 5-cycles. We show that the problem to determine whether $G$ is $(0,k)$-colorable is NP-complete for each positive integer $k.$ Moreover, we construct non-$(1,k)$-colorable planar graphs without…

We systematically investigate the complexity of counting subgraph patterns modulo fixed integers. For example, it is known that the parity of the number of $k$-matchings can be determined in polynomial time by a simple reduction to the…

Let $\alpha \in \mathbb{R}$ and let $$A=\begin{bmatrix} 1 & 1 \\ 0 & 1\end{bmatrix} \ \text{and} \ B_{\alpha} = \begin{bmatrix} 1 & 0 \\ \alpha & 1\end{bmatrix}.$$ The subgroup $G_\alpha$ of $\mathrm{SL}_2(\mathbb{R})$ is a group generated…

Let $G$ be a finite group, and let ${\rm{cd}}(G)$ denote the set of degrees of the irreducible complex characters of $G$. The degree graph $\Delta(G)$ of $G$ is defined as the simple undirected graph whose vertex set ${\rm{V}}(G)$ consists…

We study a problem motivated by a question related to quantum-error-correcting codes. Combinatorially, it involves the following graph parameter: $$f(G)=\min\set{|A|+|\{x\in V\setminus A : d_A(x)\text{is odd}\}| : A\neq\emptyset},$$ where…

We present a new model for hybrid planarity that relaxes existing hybrid representations. A graph $G = (V,E)$ is $(k,p)$-planar if $V$ can be partitioned into clusters of size at most $k$ such that $G$ admits a drawing where: (i) each…

Gallai asked in 1984 if any $k$-critical graph on $n$ vertices contains at least $n$ distinct $(k-1)$-critical subgraphs. The answer is trivial for $k\leq 3$. Improving a result of Stiebitz, Abbott and Zhou proved in 1995 that for all…

We investigate the problem of drawing two posets of the same ground set so that one is drawn from left to right and the other one is drawn from the bottom up. The input to this problem is a directed graph $G = (V, E)$ and two sets $X, Y$…

For a graph $G$, let $\chi(G)$ denote the chromatic number of $G$. Given a graph $G$, the $reconfiguration$ $graph$ $for$ $the$ $k$-$colorings$ of $G$, denoted by ${\cal R}_k(G)$, is the graph whose vertices are the $k$-colorings of $G$ and…

One of the major problems in combinatorics is to determine the number of $r$-uniform hypergraphs ($r$-graphs) on $n$ vertices which are free of certain forbidden structures. This problem dates back to the work of Erd\H{o}s, Kleitman and…