Related papers: Critical group structure from the parameters of a …
Strongly regular graphs (SRGs) are highly symmetric combinatorial objects, with connections to many areas of mathematics including finite fields, finite geometries, and number theory. One can construct an SRG via the Cayley Graph of a…
Strongly regular Cayley graphs with Paley parameters over abelian groups of rank 2 were studied in [J.A Davis, Partial difference sets in p-groups, Arch.Math.63 (1994) 103-110; K.H Leung, S.L. Ma, Partial difference sets with Paley…
We give a construction of strongly regular Cayley graphs on finite fields $\F_q$ by using union of cyclotomic classes and index 4 Gauss sums. In particular, we obtain two infinite families of strongly regular graphs with new parameters.
This paper investigates the critical group of a faithful representation of a finite group. It computes the order of the critical group in terms of the character values, and gives some restrictions on its subgroup structure. It also computes…
The Conway-99 problem questions the existence of a strongly regular graph with 99 vertices and specific parameters. A \textit{strongly} regular graph is a regular graph that exhibits two additional properties: vertices must share a fixed…
In extremal combinatorics, it is common to focus on structures that are minimal with respect to a certain property. In particular, critical and list-critical graphs occupy a prominent place in graph coloring theory. Stiebitz, Tuza, and…
We introduce the notion of q-analogs of strongly regular graphs and give several examples of such structures. We prove a necessary condition on the parameters, show the connection to designs over finite fields, and present a classification.
We expound a concise construction of finite groups and groupoids whose Cayley graphs satisfy graded acyclicity requirements. Our acyclicity criteria concern cyclic patterns formed by coset-like configurations w.r.t. subsets of the generator…
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…
We discuss the problem of finding critical sets in graphs, a concept which has appeared in a number of guises in the combinatorics and graph theory literature. The case of the Sudoku graph receives particular attention, because critical…
We prove that there is no strongly regular graph (SRG) with parameters (460,153,32,60). The proof is based on a recent lower bound on the number of 4-cliques in a SRG and some applications of Euclidean representation of SRGs.
Given feasible strongly regular graph parameters $(v,k,\lambda,\mu)$ and a non-negative integer $d$, we determine upper and lower bounds on the order of a $d$-regular induced subgraph of any strongly regular graph with parameters…
This paper is originally designed as a part of revision of the author's preprint math.AG/9908174 "P-adic Schwarzian triangle groups of Mumford type". Recently, Yves Andr'e pointed out a flaw in that preprint; more precisely, Proposition II…
In this paper, we give a new lifting construction of "hyperbolic" type of strongly regular Cayley graphs. Also we give new constructions of strongly regular Cayley graphs over the additive groups of finite fields based on partitions of…
We define three new pebbling parameters of a connected graph $G$, the $r$-, $g$-, and $u$-critical pebbling numbers. Together with the pebbling number, the optimal pebbling number, the number of vertices $n$ and the diameter $d$ of 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…
A random algebraic graph is defined by a group $G$ with a uniform distribution over it and a connection $\sigma:G\longrightarrow[0,1]$ with expectation $p,$ satisfying $\sigma(g)=\sigma(g^{-1}).$ The random graph…
We consider the Erdos-Renyi random graph G(n,p) inside the critical window, where p = 1/n + lambda * n^{-4/3} for some lambda in R. We proved in a previous paper (arXiv:0903.4730) that considering the connected components of G(n,p) as a…
Let $G$ be a finite group and $G_p$ be a Sylow $p$-subgroup of $G$ for a prime $p$ in $\pi(G)$, the set of all prime divisors of the order of $G$. The automiser $A_p(G)$ is defined to be the group $N_G(G_p)/G_pC_G(G_p)$. We define the Sylow…
Strongly regular graphs are highly symmetrical and can be described fully with just a few parameters, yet the existence of many of them is still under the question. In this paper, we continue the study of the famuly of strongly regular…