This paper presents an alternative proof of the celebrated friendship theorem, originally established by Erdős, Rényi, and Sós (1966). The proof relies on a closed-form expression for the Lovász $\vartheta$-function of strongly regular graphs, recently derived by the author. Additionally, the paper considers some known extensions of the theorem, offering discussions that provide insights into the friendship theorem, one of its extensions, and the proposed proof. Leveraging the closed-form expression for the Lovász $\vartheta$-function of strongly regular graphs, the paper further establishes new necessary conditions for a strongly regular graph to be a spanning or induced subgraph of another strongly regular graph. In the case of induced subgraphs, the analysis also incorporates a property of graph energies. Some of these results are extended to regular graphs and their subgraphs.
The quadratic embedding constant (QE constant) of a graph is a new characteristic value of a graph defined through the distance matrix. We derive formulae for the QE constants of the join of two regular graphs, double graphs and certain lexicographic product graphs. Examples include complete bipartite graphs, wheel graphs, friendship graphs, completely split graph, and some graphs associated to strongly regular graphs.
A graph $G$ with $d+1$ distinct eigenvalues is called strongly distance-regular if $G$ itself is distance-regular, and its distance-$d$ graph $G_d$ is strongly-regular. In this note we provide a spectral characterization of those distance-regular graphs with diameter $d=4$ which are strongly distance-regular. As a byproduct, it is shown that all bipartite strongly distance-regular graphs with such a diameter are antipodal.
The concept of directed strongly regular graphs was introduced by Duval in his paper, A Directed Graph Version of Strongly Regular Graphs. Duval also provided several construction methods for directed strongly regular graphs. The directed strongly regular graphs reported in this paper are obtained using a block construction of adjacency matrices of regular tournaments and circulant matrices. We then give some algebraic and combinatorial interpretation of these graphs in connection with known directed strongly regular graphs and related combinatorial structures.
Directed strongly regular graphs were introduced by Duval in 1998 as one of the possible generalization of classical strongly regular graphs to the directed case. Duval also provided several construction methods for directed strongly regular graphs. In this paper, an infinite family of directed strongly regular graphs is constructed, as generalized Cayley graphs of cyclic groups.
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 subdifference sets of the Singer difference sets. Our results unify some recent constructions of strongly regular Cayley graphs related to $m$-ovoids and $i$-tight sets in finite geometry. Furthermore, some of the strongly regular Cayley graphs obtained in this paper are new or nonisomorphic to known strongly regular graphs with the same parameters.
We prove that every connected strongly regular graph on sufficiently many vertices is Hamiltonian. We prove this by showing that, apart from three families, connected strongly regular graphs are (highly) pseudo-random. Our results suggest a number of new questions and conjectures.
A simplified version of the theory of strongly regular graphs is developed for the case in which the graphs have no triangles. This leads to (i) direct proofs of the Krein conditions, and (ii) the characterization of strongly regular graphs with no triangles such that the second subconstituent is also strongly regular. The method also provides an effective means of listing feasible parameters for such graphs.
Given feasible strongly regular graph parameters $(v,k,λ,μ)$ 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 $(v,k,λ,μ)$. Our new bounds are at least as good as the bounds on the order of a $d$-regular induced subgraph of a $k$-regular graph determined by Haemers. Further, we prove that for each non-negative integer $d$, our new upper bound improves on Haemers' upper bound for infinitely many strongly regular graphs.
The graph $G$ is said to be strongly regular with parameters $(n,k,λ,μ)$ if the following conditions hold: (1) each vertex has $k$ neighbours; (2) any two adjacent vertices of $G$ have $λ$ common neighbours; (3) any two non-adjacent vertices of $G$ have $μ$ common neighbours. In this paper we study two weaker notions of strongly regular graphs. A graph satisfying the conditions $(1)$ and $(2)$ is called an edge-regular graph with parameters $(n,k,λ)$. We call a graph satisfying the conditions $(1)$ and $(3)$ a pseudo strongly regular graph with parameters $(n,k,μ)$. In this paper we study the …
In this paper, we give a construction of strongly regular Cayley graphs on the additive groups of finite fields based on three-valued Gauss periods. As consequences, we obtain two infinite families and one sporadic example of new strongly regular Cayley graphs. This construction can be viewed as a generalization of that of strongly regular Cayley graphs obtained in \cite{BLMX}.
We study a generalization of strongly regular graphs. We call a graph strongly walk-regular if there is an $\ell >1$ such that the number of walks of length $\ell$ from a vertex to another vertex depends only on whether the two vertices are the same, adjacent, or not adjacent. We will show that a strongly walk-regular graph must be an empty graph, a complete graph, a strongly regular graph, a disjoint union of complete bipartite graphs of the same size and isolated vertices, or a regular graph with four eigenvalues. Graphs from the first three families in this list are indeed strongly $\ell$-w…
In 1988, Duval introduced the concept of directed strongly regular graphs, which can be viewed as a directed graph version of strongly regular graphs. Such directed graphs have similar structural and algebraic properties to strongly regular graphs. In the past three decades, it was found that Cayley graphs, especially those over dihedral groups, play a key role in the construction of directed strongly regular graphs. In this paper, we focus on the characterization of directed strongly regular Cayley graphs over more general groups. Let $G$ be a non-abelian group with an abelian subgroup of ind…
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 parameters.
In this paper, we generalize the construction of strongly regular graphs in [Y. Tan et al., Strongly regular graphs associated with ternary bent functions, J. Combin.Theory Ser. A (2010), 117, 668-682] from ternary bent functions to $p$-ary bent functions, where $p$ is an odd prime. We obtain strongly regular graphs with three types of parameters. Using certain non-quadratic $p$-ary bent functions, our constructions can give rise to new strongly regular graphs for small parameters.
In this paper, we construct intriguing sets in five classes of strongly regular graphs defined on nonisotropic points of finite classical polar spaces, and determine their intersection numbers.
In this paper,we construct some directed strongly regular Cayley graphs on dihedral groups,these generalizes some earlier constructions.We also characterize some certain directed strongly regular Cayley graphs on dihedral groups $D_{p^α}$,where $p$ is a prime and $α\geqslant 1$ is a positive integer.
Duval, in "A Directed Graph Version of Strongly Regular Graphs" [{\it Journal of Combinatorial Theory}, Series A 47 (1988) 71 - 100], introduced the concept of directed strongly regular graphs. In this paper we construct several rich families of directed strongly regular graphs with new parameters. Our constructions yielding new parameters are based on extending known explicit constructions to cover more parameter sets. We also explore some of the links between Cayley graphs, block matrices and directed strongly regular graphs with certain parameters. Directed strongly regular graphs which are…
In this paper, generalizing the result in \cite{GXY}, we construct strongly regular Cayley graphs by using union of cyclotomic classes of $\F_q$ and Gauss sums of index $w$, where $w\geq 2$ is even. In particular, we obtain three infinite families of strongly regular graphs with new parameters.
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.
For any positive integer $m$, the complete graph on $2^{2m}(2^m+2)$ vertices is decomposed into $2^m+1$ commuting strongly regular graphs, which give rise to a symmetric association scheme of class $2^{m+2}-2$. Furthermore, the eigenmatrices of the symmetric association schemes are determined explicitly. As an application, the eigenmatrix of the commutative strongly regular decomposition obtained from the strongly regular graphs is derived.
We consider imprimitive association schemes of class four which are two-fold covers of strongly regular graphs with spreads. It will be shown that a two-fold cover of a strongly regular graph with a spread provides a five class fission scheme of the imprimitive scheme of class four.
In this article we construct a series of new infinite families of strongly regular graphs with the same parameters as the point-graphs of non-singular quadrics in PG(n,2).
In this paper, we construct directed strongly regular graphs with new parameters by using partial sum families with local rings. 16 families of new directed strongly regular graphs are obtained and the uniform partial sum families are given. Based on the cyclotomic numbers of finite fields, we present two infinite families of directed strongly regular Cayley graphs from semi-direct products of groups.
The concept of directed strongly regular graphs (DSRG) was introduced by Duval in "A Directed Graph Version of Strongly Regular Graphs" [Journal of Combinatorial Theory, Series A 47(1988)71-100]. Duval also provided several construction methods for directed strongly regular graphs. In this paper, We construct several new classes of directed strongly regular graphs which are obtained by using Kronecker matrix product, Semidirect product and Cayley coset graph. At the same time, using group representation, for two special cases, we give some other sufficient and necessary conditions of Cayley gr…
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 geometries and group divisible designs. In this paper, we use some collections of antiflags (not the entire set of antiflags) of tactical configurations to construct another couple of infinite families of directed strongly regular graphs. Our construction methods are capable of producing many, if not all, nonisomorphic directed strongly regular graphs with sam…
We determine new upper bounds for the clique numbers of strongly regular graphs in terms of their parameters. These bounds improve on the Delsarte bound for infinitely many feasible parameter tuples for strongly regular graphs, including infinitely many parameter tuples that correspond to Paley graphs.
We show that the method of counting closed walks in strongly regular graphs rules out no parameter sets other than those ruled out by the method of counting eigenvalue multiplicities.
In 1988, D. G. Higman gave properties of parameters of strongly regular designs without proof. It included an error. We will give proofs and correct the error.
In this paper, we characterize some certain directed strongly regular Cayley graphs on Dihedral groups $D_{n}$, where $n\geqslant 3$ is a positive integer.
In this paper we unify several existing regularity conditions for graphs, including strong regularity, $k$-isoregularity, and the $t$-vertex condition. We develop an algebraic composition/decomposition theory of regularity conditions. Using our theoretical results we show that a family of non rank 3 graphs known to satisfy the $7$-vertex condition fulfills an even stronger condition, $(3,7)$-regularity (the notion is defined in the text). Derived from this family we obtain a new infinite family of non rank $3$ strongly regular graphs satisfying the $6$-vertex condition. This strengthens and ge…
A weighing matrix $W$ is quasi-balanced if $|W||W|^\top=|W|^\top|W|$ has at most two off-diagonal entries, where $|W|_{ij}=|W_{ij}|$. A quasi-balanced weighing matrix $W$ signs a strongly regular graph if $|W|$ coincides with its adjacency matrix. Among other things, signed strongly regular graphs and their equivalent association schemes are presented.
In this paper, we show that if G is strongly regular then the Gallai graph and the anti-Gallai graph of G are edge-regular. We also identify conditions under which the Gallai and anti-Gallai graphs are themselves strongly regular, as well as conditions under which they are 2-connected. We include also a number of concrete examples and a discussion of spectral properties of the Gallai and anti-Gallai graphs.
In this paper, we show that the edge connectivity of a distance-regular digraph $Γ$ with valency $k$ is $k$ and for $k>2$, any minimum edge cut of $Γ$ is the set of all edges going into (or coming out of) a single vertex. Moreover we show that the same result holds for strongly regular digraphs. These results extend the same known results for undirected case with quite different proofs.
We give two "lifting" constructions of strongly regular Cayley graphs. In the first construction we "lift" a cyclotomic strongly regular graph by using a subdifference set of the Singer difference set. The second construction uses quadratic forms over finite fields and it is a common generalization of the construction of the affine polar graphs \cite{CK86} and a construction of strongly regular Cayley graphs given in \cite{FWXY}. The two constructions are related in the following way: The second construction can be viewed as a recursive construction, and the strongly regular Cayley graphs obta…
In this paper we introduce a construction of directed strongly regular graphs from smaller ones using equitable partitions. Each equitable partition of a single DSRG satisfying several conditions leads to an infinite family of directed strongly regular graphs. We construct in this way dozens of infinite families. For order at most 110, we confirm the existence of DSRGs for 30 previously open parameter sets.
We prove the non-existence of strongly regular graph with parameters $(76,30,8,14)$. We use Euclidean representation of a strongly regular graph together with a new lower bound on the number of 4-cliques to derive strong structural properties of the graph, and then use these properties to show that the graph cannot exist.
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 generalize the concept of strong walk-regularity to directed graphs. We call a digraph strongly $\ell$-walk-regular with $\ell >1$ if the number of walks of length $\ell$ from a vertex to another vertex depends only on whether the two vertices are the same, adjacent, or not adjacent. This generalizes also the well-studied strongly regular digraphs and a problem posed by Hoffman. Our main tools are eigenvalue methods. The case that the adjacency matrix is diagonalizable with only real eigenvalues resembles the undirected case. We show that a digraph $Γ$ with only real eigenvalues whose adjac…
Over forty years ago, Goethals and Seidel showed that if the adjacency algebra of a strongly regular graph $X$ contains a Hadamard matrix then $X$ is either of Latin square type or of negative Latin square type. We extend their result to complex Hadamard matrices and find only three additional families of parameters for which the strongly regular graphs have complex Hadamard matrices in their adjacency algebras. Moreover we show that there are only three distance regular covers of the complete graph that give complex Hadamard matrices.
We show that there is no $(95,40,12,20)$ strongly regular graph and, consequently, there is no $(96,45,24,18)$ strongly regular graph, no two-graph on $96$ vertices, and no partial geometry $\rm{pg}(5,9,3)$. The main idea of the result is based on the star complement technique and requires a small amount of computation.
In this article we generalize a theorem of Benson for generalized quadrangles to strongly regular graphs and directed strongly regular graphs. The main result provides numerical restrictions on the number of fixed vertices and the number of vertices mapped to adjacent vertices under an automorphism. It is explained how these results can be used when studying partial difference sets in Abelian groups and projective two-weight sets. The underlying ideas are linear algebraic in nature.
We find new constructions of infinite families of skew Hadamard difference sets in elementary abelian groups under the assumption of the existence of cyclotomic strongly regular graphs. Our construction is based on choosing cyclotomic classes in finite fields.
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.
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 parameters (216,40,4,8) and (540,187,58,68). We also construct a strongly regular graph with parameters (540,224,88,96) that was to the best of our knowledge previously unknown. Further, we show that under certain conditions an orbit matrix $M$ of a strongly regular graph $Γ$ can be used to define a new strongly regular graph $\widetildeΓ$, where the vertices of the…
In this paper, we give a construction of strongly regular Cayley graphs and a construction of skew Hadamard difference sets. Both constructions are based on choosing cyclotomic classes in finite fields, and they generalize the constructions given by Feng and Xiang \cite{FX111,FX113}. Three infinite families of strongly regular graphs with new parameters are obtained. The main tools that we employed are index 2 Gauss sums, instead of cyclotomic numbers.
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 odd prime powers we construct a graph that is a plausible candidate for the second subconstituent. Unfortuantely we also show that the corresponding graph is strongly regular only when the prime power is 3, in which case the graph is already known.
We construct a strongly regular graph with the parameters (65; 32; 15; 16). The idea is to search for an adjacency matrix that consists of circulant blocks. Equations with such matrices can be reduced to congruences with polynomials matrices of smaller orders. We can consider these congruences over different moduli for a more efficient computational approach.
We report about the results of the application of modern computer algebra tools for construction of directed strongly regular graphs. The suggested techniques are based on the investigation of non-commutative association schemes and Cayley graphs over non-Abelian groups. We demonstrate examples of directed strongly regular graphs for 28 different parameter sets, for which the existence of a corresponding digraph has not been known before.
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 $c_{i}$ common neighbours for some $1 \leq i \leq p$. This is a generalization of a strongly regular graph. In this paper, we focus on strictly quasi-strongly regular graphs of grade $3$ with $c_i = k - i$ for $i = 1, 2, 3$. The main result is to show the sharp bounds of order $n$ for a given $k \geq 4$. Furthermore, by this result, we characterize all of these graphs…
A {\em resolving set} for a graph $Γ$ is a collection of vertices $S$, chosen so that for each vertex $v$, the list of distances from $v$ to the members of $S$ uniquely specifies $v$. The {\em metric dimension} of $Γ$ is the smallest size of a resolving set for $Γ$. A graph is {\em distance-regular} if, for any two vertices $u,v$ at each distance $i$, the number of neighbours of $v$ at each possible distance from $u$ (i.e. $i-1$, $i$ or $i+1$) depends only on the distance $i$, and not on the choice of vertices $u,v$. The class of distance-regular graphs includes all distance-transitive graphs …
We survey the area of strongly regular graphs satisfying the 4-vertex condition and find several new families. We describe a switching operation on collinearity graphs of polar spaces that produces cospectral graphs. The obtained graphs satisfy the 4-vertex condition if the original graph belongs to a symplectic polar space.
