Conference Graphs Research Articles

Bounds for the Twin-Width of Graphs

Bonnet, Kim, Thomass\'{e}, and Watrigant (2020) introduced the twin-width of a graph. We show that the twin-width of an $n$-vertex graph is less than $(n+\sqrt{n\ln n}+\sqrt{n}+2\ln n)/2$, and the twin-width of an $m$-edge graph for a positive $m$ is less than $\sqrt{3m}+ m^{1/4} \sqrt{\ln m} / (4\cdot 3^{1/4}) + 3m^{1/4} / 2$. Conference graphs of order $n$ (when such graphs exist) have twin-width at least $(n-1)/2$, and we show that Paley graphs achieve this lower bound. We also show that the twin-width of the Erd\H{o}s-R\'{e}nyi random graph $G(n,p)$ with $1/n\leq p=p(n)\leq 1/2$ is larger than $2p(1-p)n - (2\sqrt{2}+\varepsilon)\sqrt{p(1-p)n\ln n}$ asymptotically almost surely for any positive $\varepsilon$. Lastly, we calculate the twin-width of random graphs $G(n,p)$ with $p\leq c/n$ for a constant $c<1$, determining the thresholds at which the twin-width jumps from $0$ to $1$ and from $1$ to $2$.

On regular graphs equienergetic with their complements

We give necessary and sufficient conditions on the parameters of a regular graph Γ (with or without loops) such that . We study complementary equienergetic cubic graphs obtaining classifications up to isomorphisms for connected cubic graphs with single loops (5 non-isospectral pairs) and connected integral cubic graphs without loops ( or ). Then we show that, up to complements, the only bipartite regular graphs equienergetic and non-isospectral with their complements are the crown graphs or . Next, for the family of strongly regular graphs Γ we characterize all possible parameters such that . Furthermore, using this, we prove that a strongly regular graph is equienergetic to its complement if and only if it is either a conference graph or else it is a pseudo Latin square graph (i.e. has OA parameters). We also characterize all complementary equienergetic pairs of graphs of type , and in Cameron's hierarchy (the cases in the non-bipartite case and are still open). Finally, we consider unitary Cayley graphs over rings . We show that if R is a finite Artinian ring with an even number of local factors, then is complementary equienergetic if and only if is the product of 2 finite fields.

Parametric restrictions on quasi-symmetric designs

In this paper, we attach several new invariants to connected strongly regular graphs (excepting conference graphs on non-square number of vertices): one invariant called the discriminant, and a p-adic invariant corresponding to each prime number p. We prove parametric restrictions on quasi-symmetric 2-designs with a given connected block graph G and a given defect (absolute difference of the two intersection numbers) solely in terms of the defect and the parameters of G, including these new invariants. This is a natural analogue of Schutzenberger’s Theorem and the Shrikhande–Chowla–Ryser theorem. This theorem is effective when these graph invariants can be explicitly computed. We do this for complete multipartite graphs, co-triangular graphs, symplectic non-orthogonality graphs (over the field of order 2) and the Steiner graphs, yielding explicit restrictions on the parameters of quasi-symmetric 2-designs whose block graphs belong to any of these four classes.

New constructions of strongly regular Cayley graphs on abelian non p-groups

Davis and Jedwab (1997) established a great construction theory unifying many previously known constructions of difference sets, relative difference sets and divisible difference sets. They introduced the concept of building blocks, which played an important role in the theory. On the other hand, Polhill (2010) gave a construction of Paley type partial difference sets (conference graphs) based on a special system of building blocks, called a covering extended building set, and proved that there exists a Paley type partial difference set in an abelian group of order 9iv4 for any odd positive integer v>1 and any i=0,1. His result covers all orders of abelian non p-groups in which Paley type partial difference sets exist. In this paper, we give new constructions of strongly regular Cayley graphs on abelian groups by extending the theory of building blocks. The constructions are large generalizations of Polhill's construction. In particular, we show that for a positive integer m and elementary abelian groups Gi, i=1,2,…,s, of order qi4 such that 2m|qi+1, there exists a decomposition of the complete graph on the abelian group G=G1×G2×⋯×Gs by strongly regular Cayley graphs with negative Latin square type parameters (u2,c(u+1),−u+c2+3c,c2+c), where u=q12q22⋯qs2 and c=(u−1)/m. Such strongly regular decompositions were previously known only when m=2 or G is a p-group. Moreover, we find one more new infinite family of decompositions of the complete graphs by Latin square type strongly regular Cayley graphs. Thus, we obtain many strongly regular graphs with new parameters.

Bordered Complex Hadamard Matrices and Strongly Regular Graphs

We consider bordered complex Hadamard matrices whose core is contained in the Bose-Mesner algebra of a strongly regular graph. Examples include a complex Hadamard matrix whose core is contained in the Bose-Mesner algebra of a conference graph due to J. Wallis, F. Szollősi, and a family of Hadamard matrices given by Singh and Dubey. In this paper, we prove that there are no other bordered complex Hadamard matrices whose core is contained in the Bose-Mesner algebra of a strongly regular graph.

A Note on Siamese Twin Designs Intersecting in a BIBD and a PBD

Suppose there exists a Hadamard 2- $$(m,\frac{m-1}{2},\frac{m-3}{4})$$ design with skew incidence matrix, and a conference graph with v vertices, where $$v = 2m-1$$ . Under this assumption we prove that there exists a Siamese twin Menon design with parameters $$(4m^{2},2m^{2}-m,m^{2}-m)$$ intersecting in a balanced incomplete block design $$\mathrm {BIBD}(2m^{2} - m, m^{2} - m, m^{2} - m - 1)$$ and a pairwise balanced design $$\mathrm {PBD}(2m^{2} - m, \{m^{2}, m^{2} - m\}, m^{2} - m - 1)$$ . These Menon designs lead to regular amicable Hadamard matrices of orders not previously constructed. Further we construct complex orthogonal designs of order $$4m^2$$ and Butson Hadamard matrices $$\mathrm {BH}(4m^{2},2k)$$ for all k. Some results regarding automorphisms of the constructed Menon designs are proven.

Two-graphs and NSSDs: An algebraic approach

A two-graph (V,Δ) is a combinatorial entity consisting of a set V together with a collection Δ of unordered triples of elements of V, such that there exists a graph G with vertex set V in which the triples in Δ are precisely the subgraphs K3 or K2∪̇K1 induced in G. Switching is an equivalence relation partitioning the graphs on n vertices into switching classes. All the graphs that are switching equivalent to G yield the same Δ and therefore the two–graph (V,Δ) is taken to consist of the graphs in the switching class of G. The graphs in a switching class have similar (0,±1)-Seidel matrices. So the spectrum of (V,Δ) is taken as the Seidel spectrum of a graph in the switching class. The two graphs with exactly two distinct Seidel eigenvalues μ1,μ2 are regular. We show how an involution M(μ1,μ2) provides a simple way to determine structural and combinatorial properties of the graphs of a regular two-graph. If μ1+μ2=0, the regular two-graph consists of conference graphs. We show that M(μ1,−μ1) is an NSSD (non-singular graph with a singular deck) with the special property of being a nutful graph. The rich properties of a nutful NSSD reveal new spectral properties of conference graphs.

On switching classes of graphs

A class of non-oriented simple graphs is called Seidel switching self-complementary (s.s.c. for short) if the complement of any representing graph is in the same equivalence class. Relating the Seidel adjacency matrix of a graph with a Gram matrix, we introduce the 3-signature (s,t) of a switching class of n-vertex graphs. The numbers s and t are the numbers of positive and negative triples within any representing graph of the class. It appears that, for any s.s.c. class of n-vertex graphs, these numbers are equal, yielding (n3) even. Consequently if n≡3(mod4) then there is no s.s.c. class of n-vertex graphs. We also prove that this 3-signature depends only on the spectrum of the adjacency matrix of the graph. We then consider the switching classes of Paley conference graphs with 4k+2 vertices, 4k+1=pα, p an odd prime and α a positive integer. We reprove that these classes are s.s.c. Moreover, it is proven that all 4k-vertex graphs contained in a (4k+2)-vertex Paley conference graph are switching equivalent and their class is still a s.s.c. class. In addition, the 3-signature is generalized in view of obtaining a complete invariant of switching classes up to order 8.

Regular Hadamard matrices constructed from Hadamard 2-designs and conference graphs

Suppose there exists a Hadamard 2-(m,m−12,m−34) design having skew incidence matrix. If there exists a conference graph on 2m−1 vertices, then there exists a regular Hadamard matrix of order 4m2. A conference graph on 2m+3 vertices yields a regular Hadamard matrix of order 4(m+1)2.

The number of ideals of [formula omitted] containing x(x − α)(x − β) with given index

It is well-known that a connected regular graph is strongly-regular if and only if its adjacency matrix has exactly three eigenvalues. Let B denote an integral square matrix and 〈B〉 denote the subring of the full matrix ring generated by B. Then 〈B〉 is a free Z-module of finite rank, which guarantees that there are only finitely many ideals of 〈B〉 with given finite index. Thus, the formal Dirichlet series ζ〈B〉(s)=∑n≥1ann−s is well-defined where an is the number of ideals of 〈B〉 with index n. In this article we aim to find an explicit form of ζ〈B〉(s) when B has exactly three eigenvalues all of which are integral, e.g., the adjacency matrix of a strongly-regular graph which is not a conference graph with a non-squared number of vertices. By isomorphism theorem for rings, 〈B〉 is isomorphic to Z[x]/m(x)Z[x] where m(x) is the minimal polynomial of B over Q, and Z[x]/m(x)Z[x] is isomorphic to Z[x]/m(x+γ)Z[x] for each γ∈Z. Thus, the problem is reduced to counting the number of ideals of Z[x]/x(x−α)(x−β)Z[x] with given finite index where 0,α and β are distinct integers.

Beyond graph energy: Norms of graphs and matrices

In 1978 Gutman introduced the energy of a graph as the sum of the absolute values of graph eigenvalues, and ever since then graph energy has been intensively studied.Since graph energy is the trace norm of the adjacency matrix, matrix norms provide a natural background for its study. Thus, this paper surveys research on matrix norms that aims to expand and advance the study of graph energy.The focus is exclusively on the Ky Fan and the Schatten norms, both generalizing and enriching the trace norm. As it turns out, the study of extremal properties of these norms leads to numerous analytic problems with deep roots in combinatorics.The survey brings to the fore the exceptional role of Hadamard matrices, conference matrices, and conference graphs in matrix norms. In addition, a vast new matrix class is studied, a relaxation of symmetric Hadamard matrices.The survey presents solutions to just a fraction of a larger body of similar problems bonding analysis to combinatorics. Thus, open problems and questions are raised to outline topics for further investigation.

A short note on a short remark of Graham and Lovász

Let D be the distance matrix of a connected graph G and let n−(G),n+(G) be the numbers of strictly negative and positive eigenvalues of D respectively. It was remarked in Graham and Lovász (1978) that it is not known whether there is a graph for which n+(G)>n−(G). In this note we show that there exist an infinite number of graphs satisfying the stated inequality, namely the conference graphs of order >9, a major representative of this class being the Paley graphs. The result is obtained by deriving the eigenvalues of the distance matrix of a strongly regular graph.

Graphs whose normalized Laplacian has three eigenvalues

We give a combinatorial characterization of graphs whose normalized Laplacian has three distinct eigenvalues. Strongly regular graphs and complete bipartite graphs are examples of such graphs, but we also construct more exotic families of examples from conference graphs, projective planes, and certain quasi-symmetric designs.

Some characterizations of strongly regular graphs with respect to their vertex deleted subgraphs

We say that a regular graph G of order n and degree r ≥ 1 (which is not the complete graph) is strongly regular if there exist non-negative integers τ and θ such that | S i ∩ S j | = τ for any two adjacent vertices i and j , and | S i ∩ S j | = θ for any two distinct non-adjacent vertices i and j , where S k denotes the neighborhood of the vertex k . Let i be a fixed vertex from the vertex set V ( G ) = { 1 , 2 , … , n } and let G i = G ∖︀ i be its vertex deleted subgraph. Let H i be switching equivalent to G i with respect to S i ⊆ V ( G i ) . We prove that H i is a strongly regular graph of order n − 1 and degree 2 ( r − θ ) with τ ( H i ) = r + τ − 2 θ and θ ( H i ) = r − θ if and only if n = 4 r − 2 τ − 2 θ . Otherwise, we prove that H i has exactly two main eigenvalues μ 1 and μ 2 . In this case, the main eigenvalues of H i are represented in the form μ 1 , 2 = τ − θ + r ± ( τ − θ + r ) 2 + 4 ( n − 2 r ) ( r − θ ) 2 . We also prove that G i and H i are cospectral if and only if r = 2 θ . Finally, we show that if G is a strongly regular graph of order 2 ( 2 k 2 + 2 k + 1 ) and degree k ( 2 k + 1 ) with τ = k 2 − 1 and θ = k 2 then H i is a conference graph of order ( 2 k + 1 ) 2 and degree 2 k ( k + 1 ) with τ ( H i ) = k 2 + k − 1 and θ ( H i ) = k ( k + 1 ) . However, if G is a strongly regular graph of order 4 ( 3 k 2 + 3 k + 1 ) and degree 3 k ( 2 k + 1 ) with τ = 3 k 2 − k − 2 and θ = k ( 3 k + 1 ) then H i is a strongly regular graph (non-conference) of order 3 ( 2 k + 1 ) 2 and degree 2 k ( 3 k + 2 ) with τ ( H i ) = 3 k 2 − 2 and θ ( H i ) = k ( 3 k + 2 ) .

Paley type partial difference sets in non p-groups

By modifying a construction for Hadamard (Menon) difference sets we construct two infinite families of negative Latin square type partial difference sets in groups of the form $${\mathbb {Z}_3^2 \times \mathbb {Z}_p^{4t}}$$ where p is any odd prime. One of these families has the well-known Paley parameters, which had previously only been constructed in p-groups. This provides new constructions of Hadamard matrices and implies the existence of many new strongly regular graphs including some that are conference graphs. As a corollary, we are able to construct Paley---Hadamard difference sets of the Stanton-Sprott family in groups of the form $${\mathbb {Z}_3^2 \times \mathbb {Z}_p^{4t} \times {\it EA} (9p^{4t} \pm 2)}$$ when $${9p^{4t} \pm 2}$$ is a prime power. These are new parameters for such difference sets.

Some characterizations of strongly regular graphs

Let G be a connected strongly regular graph of order n and degree r. Let λ1=r, λ2 and λ3 denote the distinct eigenvalues of G. We demonstrate that the Krein parameters of the strongly regular graph G can be expressed in terms of its eigenvalues r, λ2 and λ3. Due to this fact, we prove that there exists no strongly regular graph of order n=4k+3, provided that n is a prime number. Besides, we prove that the only strongly regular graphs of order n=4k+1 (n is a prime number) are conference graphs. In this work we also establish a characterization of strongly regular graphs of order n=2(2k+1), where 2k+1 is a prime number.

Type-II Matrices Attached to Conference Graphs

We determine the Nomura algebras of the type-II matrices belonging to the Bose-Mesner algebra of a conference graph.

ON A THEOREM OF JAEGER

We give a short and alternative proof of a theorem of F. Jaeger that except for Potts models attached to the complete graphs, the only spin models associated with symmetric conference graphs with n ≥ 5 vertices are the pentagon and the lattice graph L2(3) with 9 vertices. The proof avoids Jaeger's use of the classification of strongly regular graphs having strongly regular subconstituents due to P. J. Cameron, J. M. Goethals, and J. J. Seidel.

Distance matrices, dimension, and conference graphs

We define the dimension of a distance matrix and its associated metric space, and use this to give necessary and sufficient conditions for a metric space to be isometrically embeddable into suitable real inner product spaces and Euclidean spheres. Also, for certain distance matrices C with irrational entries, we derive the bound w ≤ 2 f+ 1 for the size w of C in terms of its dimension f. This result is applied to improve a bound by Larman, Rogers, and Seidel on two-distance sets in Euclidean space, and to characterize certain regular graphs as conference graphs.