Related papers: Strongly regular graphs with parameters (81,30,9,1…

A result of Erd\"os and R\'enyi shows that for a fixed integer n almost all graphs satisfy the n-e.c. adjacency property. However, there are few explicit constructions of n e.c. graphs for n > 2, and almost all known families of n-e.c.…

We exhibit a new construction of edge-regular graphs with regular cliques that are not strongly regular. The infinite family of graphs resulting from this construction includes an edge-regular graph with parameters $(24,8,2)$. We also show…

A k-regular graph on v vertices is a divisible design graph with parameters (v, k, lambda_1 ,lambda_2, m, n) if its vertex set can be partitioned into m classes of size n, such that any two different vertices from the same class have…

A graph is \textit{rigid} if it only admits the identity endomorphism. We show that for every $d\ge 3$ there exist infinitely many mutually rigid $d$-regular graphs of arbitrary odd girth $g\geq 7$. Moreover, we determine the minimum order…

A graph $G$ is a link-irregular graph if every two distinct vertices of $G$ have non-isomorphic links. The link of a vertex $v$ in $G$ is the subgraph induced by the neighbors of $v$ in $G$. Ali, Chartrand and Zhang [Discussiones…

A half-square of a bipartite graph $B=(X,Y,E_B)$ has one color class of $B$ as vertex set, say $X$; two vertices are adjacent whenever they have a common neighbor in $Y$. If $G=(V,E_G)$ is the half-square of a planar bipartite graph…

The graph $G$ is said to be strongly regular with parameters $(n,k,\lambda,\mu)$ if the following conditions hold: (1) each vertex has $k$ neighbours; (2) any two adjacent vertices of $G$ have $\lambda$ common neighbours; (3) any two…

Twin-width is a recently introduced graph parameter based on the repeated contraction of near-twins. It has shown remarkable utility in algorithmic and structural graph theory, as well as in finite model theory -- particularly since…

Praeger-Xu graphs are connected, symmetric, 4-regular graphs that are unusual both in that their automorphism groups are large, and in that vertex stabilizer subgroups are also large. Determining number and distinguishing number are…

In 2018 the first, Rukavina and the third author constructed with the aid of a computer the first example of a strongly regular graph $\Gamma$ with parameters (216, 40, 4, 8) and proved that it is the unique PSU(4,2)-invariant…

We give two constructions of strongly regular Cayley graphs on finite fields $\F_q$ by using union of cyclotomic classes and index 2 Gauss sums. In particular, we obtain twelve infinite families of strongly regular graphs with new…

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 introduce a new notation for representing labeled regular bipartite graphs of arbitrary degree. Several enumeration problems for labeled and unlabeled regular bipartite graphs have been introduced. A general algorithm for enumerating all…

A quasi-strongly regular graph of grade $p$ with parameters $(n, k, a; c_1, \ldots, c_p)$ is a $k$-regular graph of order $n$ such that any two adjacent vertices share $a$ common neighbours and any two non-adjacent vertices share $c_{i}$…

The symplectic graph Sp(2d, q) is the collinearity graph of the symplectic space of dimension 2d over a finite field of order q. A k-regular graph on v vertices is a divisible design graph with parameters (v, k, lambda_1, lambda_2 ,m,n) if…

For every integer d > 9, we construct infinite families {G_n}_n of d+1-regular graphs which have a large girth > log_d |G_n|, and for d large enough > 1,33 log_d |G_n|. These are Cayley graphs on PGL_2(q) for a special set of d+1 generators…

In the $m$-dimensional affine space $AG(m,q)$ over the finite field $\mathbb{F}_q$ of odd order $q$, the analogous of the Euclidean distance gives rise to a graph $\mathfrak{G}_{m,q}$ where vertices are the points of $AG(m,q)$ and two…

This article investigates the isomorphism problem for graphs derived from the four standard graph products: Cartesian, Kronecker (direct), strong, and lexicographic product. We provide a complete characterization of all simple connected…

We describe a general construction of strongly regular graphs from the collinearity graph of a finite classical polar spaces of rank at least $3$ over a finite field of order $q$. We show that these graphs are non-isomorphic to the…

The existence of $srg(99,14,1,2)$ has been a question of interest for several decades to the moment. In this paper we consider the structural properties in general for the family of strongly regular graphs with parameters $\lambda =1$ and…