Edge-regular Graphs Research Articles

On the Existence of Small Strictly Neumaier Graphs

A Neumaier graph is a non-complete edge-regular graph containing a regular clique. In this work, we prove several results on the existence of small strictly Neumaier graphs. In particular, we present a theoretical proof of the uniqueness of the smallest strictly Neumaier graph with parameters (16, 9, 4; 2, 4), we establish the existence of a strictly Neumaier graph with parameters (25, 12, 5; 2, 5), and we disprove the existence of strictly Neumaier graphs with parameters (25, 16, 9; 3, 5), (28, 18, 11; 4, 7), (33, 24, 17; 6, 9), (35, 2212; 3, 5), (40, 30, 22; 7, 10) and (55, 34, 18; 3, 5). Our proofs use combinatorial techniques and a novel application of integer programming methods.

On Graphs in Which the Neighborhoods of Vertices Are Edge-Regular Graphs without 3-Claws

On Graphs in Which the Neighborhoods of Vertices Are Edge-Regular Graphs without 3-Claws

An infinite class of Neumaier graphs and non-existence results

A Neumaier graph is a non-complete edge-regular graph containing a regular clique. A Neumaier graph that is not strongly regular is called a strictly Neumaier graph. In this work we present a new construction of strictly Neumaier graphs, and using Jacobi sums, we show that our construction produces infinitely many instances. Moreover, we prove some necessary conditions for the existence of (strictly) Neumaier graphs that allow us to show that several parameter sets are not admissible.

Density results for perfectly regular and perfectly edge-regular fuzzy graphs

This paper studies the fuzzy graph properties of density and being self-complimentary in perfectly regular and perfectly edge-regular fuzzy graphs. Several structural results for these classes of fuzzy graphs are provided. These results are parameterized by the parameters which define perfectly regular and perfectly edge-regular fuzzy graphs, thus illustrating the computational benefits of using these classes of fuzzy graphs to represent and/or approximate fuzzy networks when performing analyses of fuzzy networks.

Neumaier graphs with few eigenvalues

A Neumaier graph is a non-complete edge-regular graph containing a regular clique. In this paper we give some sufficient and necessary conditions for a Neumaier graph to be strongly regular. Further we show that there does not exist Neumaier graphs with exactly four distinct eigenvalues. We also determine the Neumaier graphs with smallest eigenvalue -2.

A novel algorithm for generating an edge-regular graph

Most recent research in graph theory considered the studying of edge regular graph class ER(n,d,λ) for some n>d>λ, such that all the n vertices have d adjacent neighbors, and each pair of adjacent vertices have λ common neighbors, none of these research gave an algorithm to generate this class with considering the λ, in this paper, a novel algorithm to generate random edge-regular graphs for λ > 0. The algorithm produces an edge-regular graph for different values of d and n Moreover, it is easy to implement and quite fast in practice. The Matlab programming language used to obtain the experimental result. The analysis of this method showed that the running time of this method proven to be O(d3).

A Novel Decision-Making Approach under Complex Pythagorean Fuzzy Environment

A complex Pythagorean fuzzy set (CPFS) is an extension of a Pythagorean fuzzy set that is used to handle the vagueness with the degrees whose ranges are enlarged from real to complex subset with unit disc. In this research study, we propose the innovative concept of complex Pythagorean fuzzy graphs (CPFGs). Further, we present the concepts of regular and edge regular graphs in a complex Pythagorean fuzzy environment. Moreover, we develop a complex Pythagorean fuzzy graph based multi-attribute decision making an approach to handling the situations in which the graphic structure of attributes is obscure. A numerical example concerning information technology improvement project selection is utilized to illustrate the availability of the developed approach.

The Smallest Strictly Neumaier Graph and its Generalisations

A regular clique in a regular graph is a clique such that every vertex outside of the clique is adjacent to the same positive number of vertices inside the clique. We continue the study of regular cliques in edge-regular graphs initiated by A. Neumaier in the 1980s and attracting current interest. We thus define a Neumaier graph to be an non-complete edge-regular graph containing a regular clique, and a strictly Neumaier graph to be a non-strongly regular Neumaier graph. We first prove some general results on Neumaier graphs and their feasible parameter tuples. We then apply these results to determine the smallest strictly Neumaier graph, which has $16$ vertices. Next we find the parameter tuples for all strictly Neumaier graphs having at most $24$ vertices. Finally, we give two sequences of graphs, each with $i^{th}$ element a strictly Neumaier graph containing a $2^{i}$-regular clique (where $i$ is a positive integer) and having parameters of an affine polar graph as an edge-regular graph. This answers questions recently posed by G. Greaves and J. Koolen.

The Classification of $2$-Extendable Edge-Regular Graphs with Diameter $2$

Let $\ell$ denote a non-negative integer. A connected graph $\Gamma$ of even order at least $2\ell+2$ is $\ell$-extendable if it contains a matching of size $\ell$ and if every such matching is contained in a perfect matching of $\Gamma$. A connected regular graph $\Gamma$ is edge-regular, if there exists an integer $\lambda$ such that every pair of adjacent vertices of $\Gamma$ have exactly $\lambda$ common neighbours. In this paper we classify $2$-extendable edge-regular graphs of even order and diameter $2$.

Another construction of edge-regular graphs with regular cliques

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 that edge-regular graphs with 1-regular cliques that are not strongly regular must have at least 24 vertices.

Edge-girth-regular graphs

We consider a new type of regularity we call edge-girth-regularity. An edge-girth-regular (v,k,g,λ)-graph Γ is a k-regular graph of order v and girth g in which every edge is contained in λ distinct g-cycles. This concept is a generalization of the well-known concept of (v,k,λ)-edge-regular graphs (that count the number of triangles) and appears in several related problems such as Moore graphs and Cage and Degree/Diameter Problems. All edge- and arc-transitive graphs are edge-girth-regular as well. We derive a number of basic properties of edge-girth-regular graphs, systematically consider cubic and tetravalent graphs from this class, and introduce several constructions that produce infinite families of edge-girth-regular graphs. We also exhibit several surprising connections to regular embeddings of graphs in orientable surfaces.

Edge-regular graphs with regular cliques

We exhibit infinitely many examples of edge-regular graphs that have regular cliques and that are not strongly regular. This answers a question of Neumaier from 1981.

Regular clique assemblies, configurations, and friendship in Edge-Regular graphs

An edge-regular graph is a regular graph in which, for some $\lambda$, any two adjacent vertices have exactly $\lambda$ common neighbors. This paper is about the existence and structure of edge-regular graphs with $\lambda =1$ and about edge-regular graphs with $\lambda >1$ which have local neighborhood structure analogous to that of the edge-regular graphs with $\lambda =1$.

On cliques in edge-regular graphs

Let Γ be an edge-regular graph with given parameters (v,k,λ). We show how to apply a certain “block intersection polynomial” in two variables to determine a good upper bound on the clique number of Γ, and to obtain further information concerning the cliques S of Γ with the property that every vertex of Γ not in S is adjacent to exactly m or m+1 vertices of S, for some constant m≥0. Some interesting examples are studied using computation with groups and graphs.

On strongly regular graphs with b1 < 26

Let Γ be a connected edge-regular graph with parameters (v, k, λ), b 1 = k−λ−1. It is well known that if b 1 = 1, then Γ is a polygon or a complete multipartite graph with colour classes of order 2. The classification of graphs with b 1 ≤ 4 is available. Even in the case b 1 = 5 the study of graphs offers great difficulties. However, the situation is much easier for strongly regular graphs. In the present paper we classify strongly regular graphs with b 1 < 26.

On strongly regular graphs with b 1 < 24

Let γ be a connected edge-regular graph with parameters (v, k, λ), and let b1 = k−λ−1. It is well known that, if b1 = 1, then Γ is either a polygon or a complete multipartite graph with parts of order 2. Graphs with b1 ≤ 4 were classified earlier. The investigation of graphs even in the case b1 = 5 involves great difficulties. However, for strongly regular graphs, the situation is much simpler. In this paper, we classify strongly regular graphs with b1 < 24.

On graphs whose local subgraphs are strongly regular with parameters (115, 18, 1, 3)

We consider undirected graphs without loops or multiple edges. If a and b are vertices in a graph Γ, then d(a, b) denotes the distance between a and b, and Γi(a) denotes the subgraph of Γ induced by the set of vertices of Γ that are a distance of i away from a. The subgraph Γ1(a) is called the neighborhood of a and is denoted by (a). Γ is called a regular graph of degree k if (a) contains precisely k vertices for any vertex a in Γ. Γ is called an edgeregular graph with parameters ( v, k, λ) if Γ is a regular graph of degree k on v vertices and each edge in Γ lies in λ triangles. Γ is said to be an amply regular graph with parameters (v, k, λ, μ) if Γ is an edgereg� ular graph with the corresponding parameters and the subgraph (a) ∩ (b) contains μ vertices in the case of d(a, b) = 2. An amply regular graph of diameter 2 is called a strongly regular graph. A graph Γ of diameter d is said to be antipodal if the relation of coincidence or being a distance of d apart on its vertex set is an equivalence relation. An antipo� dal quotient Γ' is a graph whose vertices are the antip� odal classes of Γ and two classes are adjacent if a vertex of one class is adjacent to a vertex of the other class. An antipodal graph Γ is called an rcovering (of its antip� odal quotient) if each of its antipodal classes contains precisely r vertices. If vertices u and w are separated by a distance of i in Γ, then bi(u, w) (ci(u, w)) denotes the number of vertices in the intersection of Γi +1 (u) (Γi -1 (u)) with (w). A graph Γ of diameter d is called a distanceregular graph with the intersection array {b 0 , b 1 , …, b d -1 ; c 1 , c2, …, cd} if the values bi(u, w) and ci(u, w) are indepen� dent of the choice of the vertices u and w separated by a distance of i. Let ai = k - bi - ci.

Edge Regular Graph Products

A regular nonempty graph $\Gamma$ is called edge regular, whenever there exists a nonegative integer $\lambda_{\Gamma}$, such that any two adjacent vertices of $\Gamma$ have precisely $\lambda_{\Gamma}$ common neighbours. An edge regular graph $\Gamma$ with at least one pair of vertices at distance 2 is called amply regular, whenever there exists a nonegative integer $\mu_{\Gamma}$, such that any two vertices at distance 2 have precisely $\mu_{\Gamma}$ common neighbours. In this paper we classify edge regular graphs, which can be obtained as a strong product, or a lexicographic product, or a deleted lexicographic product, or a co-normal product of two graphs. As a corollary we determine which of these graphs are amply regular.

On the commutation graph of cyclic TI-subgroups in linear groups

We study the commutation graph Γ(A) of a cyclic TI-subgroup A of order 4 in a finite group G with quasisimple generalized Fitting subgroup F*(G). It is proved that, if F*(G) is a linear group, then the graph Γ(A) is either a coclique or an edge-regular graph but not a coedge-regular graph.

On almost good triples of vertices in edge regular graphs

Consider a connected edge regular graph Γ with parameters (v, k, λ) and put b1 = k−λ−1. A triple (u, w, z) of vertices is called (almost) good whenever d(u, w) = d(u, z) = 2 and µ(u, w)+µ(u, z) ≤ 2k − 4b1 + 3 (and µ(u, w) + µ(u, z) = 2k − 4b1 + 4). If k = 3b1 + γ with γ ≥ −2, a triple (u, w, z) is almost good, and Δ = [u] ∩ [w] ∩ [z] then: either |Δ| ≤ 2; or Δ is a 3-clique and Γ is a Clebsch graph; or Δ is a 3-clique, k = 16, b1 = 6, and v = 31; or Δ is a 4-clique and Γ is a Schlafli graph.