Godsil-McKay Switching Research Articles

Constructing the Mate of Cospectral 5-regular Graphs with and without a Perfect Matching

The problem of finding a perfect matching in an arbitrary simple graph is well known and popular in graph theory. It is used in various fields, such as chemistry, combinatorics, game theory etc. The matching of M in a simple graph G is a set of pairwise nonadjacent edges, ie, those that do not have common vertices. Matching is called perfect if it covers all vertices of the graph, ie each of the vertices of the graph is incidental to exactly one of the edges. By Koenig's theorem, regular bipartite graphs of positive degree always have perfect matching. However, graphs that are not bipartite need further research.
 Another interesting problem of graph theory is the search for pairwise nonisomorphic cospectral graphs. In addition, it is interesting to find cospectral graphs that have additional properties. For example, finding cospectral graphs with and without a perfect matching.
 The fact that for each there is a pair of cospectral connected k-regular graphs with and without a perfect matching had been investigated by Blazsik, Cummings and Haemers. The pair of cospectral connected 5-regular graphs with and without a perfect matching is constructed by using Godsil-McKay switching in the paper.

Godsil-McKay switching for mixed and gain graphs over the circle group

In this paper we describe two methods, both inspired from Godsil-McKay switching on simple graphs, to build cospectral gain graphs whose gain group consists of the complex numbers of modulus 1 (the circle group). The results obtained here can be also applied to the Hermitian matrix of mixed graphs.

Cospectral Bipartite Graphs with the Same Degree Sequences but with Different Number of Large Cycles

Finding the multiplicity of cycles in bipartite graphs is a fundamental problem of interest in many fields including the analysis and design of low-density parity-check (LDPC) codes. Recently, Blake and Lin computed the number of shortest cycles (g-cycles, where g is the girth of the graph) in a bi-regular bipartite graph, in terms of the degree sequences and the spectrum (eigenvalues of the adjacency matrix) of the graph (Blake and Lin in IEEE Trans Inf Theory 64(10): 6526–6535, 2018). This result was subsequently extended in Dehghan and Banihashemi (IEEE Trans Inf Theory 65(6):3778–3789, 2019) to cycles of length $$g+2, \ldots , 2g-2$$, in bi-regular bipartite graphs, as well as 4-cycles and 6-cycles in irregular and half-regular bipartite graphs, with $$g \ge 4$$ and $$g \ge 6$$, respectively. In this paper, we complement these positive results with negative results demonstrating that the information of the degree sequences and the spectrum of a bipartite graph is, in general, insufficient to count (a) the i-cycles, $$i \ge 2g$$, in bi-regular graphs, (b) the i-cycles for any $$i > g$$, regardless of the value of g, and g-cycles for $$g \ge 6$$, in irregular graphs, and (c) the i-cycles for any $$i > g$$, regardless of the value of g, and g-cycles for $$g \ge 8$$, in half-regular graphs. To obtain these results, we construct counter-examples using the Godsil–McKay switching.

New strongly regular graphs from finite geometries via switching

We show that the strongly regular graph on non-isotropic points of one type of the polar spaces of type U(n,2), O(n,3), O(n,5), O+(n,3), and O−(n,3) are not determined by its parameters for n≥6. We prove this by using a variation of Godsil-McKay switching recently described by Wang, Qiu, and Hu. This also results in a new, shorter proof of a previous result of the first author which showed that the collinearity graph of a polar space is not determined by its spectrum. The same switching gives a linear algebra explanation for the construction of a large number of non-isomorphic designs.

Graph switching, 2-ranks, and graphical Hadamard matrices

We study the behavior of the 2-rank of the adjacency matrix of a graph under Seidel and Godsil–McKay switching, and apply the result to graphs coming from graphical Hadamard matrices of order 4m. Starting with graphs from known Hadamard matrices of order 64, we find (by computer) many Godsil–McKay switching sets that increase the 2-rank. Thus we find strongly regular graphs with parameters (63,32,16,16), (64,36,20,20), and (64,28,12,12) for almost all feasible 2-ranks. In addition we work out the behavior of the 2-rank for a graph product related to the Kronecker product for Hadamard matrices, which enables us to find many graphical Hadamard matrices of order 4m for which the number of related strongly regular graphs with different 2-ranks is unbounded as a function of m. The paper extends results from the article ‘Switched symplectic graphs and their 2-ranks’ by the first and the last author.

Isospectral discrete and quantum graphs with the same flip counts and nodal counts

The existence of non-isomorphic graphs which share the same Laplace spectrum (to be referred to as isospectral graphs) leads naturally to the following question: what additional information is required in order to resolve isospectral graphs? It was suggested by Band, Shapira and Smilansky that this might be achieved by either counting the number of nodal domains or the number of times the eigenfunctions change sign (the so-called flip count) (Band et al 2006 J. Phys. A: Math. Gen. 39 13999–4014; Band and Smilansky 2007 Eur. Phys. J. Spec. Top. 145 171–9). Recent examples of (discrete) isospectral graphs with the same flip count and nodal count have been constructed by Ammann by utilising Godsil–McKay switching (Ammann private communication). Here, we provide a simple alternative mechanism that produces systematic examples of both discrete and quantum isospectral graphs with the same flip and nodal counts.

Cospectral mates for the union of some classes in the Johnson association scheme

Let n≥k≥2 be two integers and S a subset of {0,1,…,k−1}. The graph JS(n,k) has as vertices the k-subsets of the n-set [n]={1,…,n} and two k-subsets A and B are adjacent if |A∩B|∈S. In this paper, we use Godsil–McKay switching to prove that for m≥0, k≥max⁡(m+2,3) and S={0,1,...,m}, the graphs JS(3k−2m−1,k) are not determined by spectrum and for m≥2, n≥4m+2 and S={0,1,...,m} the graphs JS(n,2m+1) are not determined by spectrum. We also report some computational searches for Godsil–McKay switching sets in the union of classes in the Johnson scheme for k≤5.

Constructing cospectral graphs via a new form of graph product

We present a new method to construct a family of co-spectral graphs. Our method is based on a new type of graph product that we define, the bipartite graph product, which may be of self-interest. Our method is different from existing techniques in the sense that it is not based on a sequence of local graph operations (e.g. Godsil–McKay switching). The explicit nature of our construction allows us, for example, to construct an infinite family of cospectral graphs and provide an easy proof of non-isomorphism. We are also able to characterize fully the spectrum of the cospectral graphs.

Unification of Graph Products and Compatibility with Switching

We define the type of graph products, which enable us to treat many graph products in a unified manner. These unified graph products are shown to be compatible with Godsil–McKay switching. Furthermore, by this compatibility, we show that the Doob graphs can also be obtained from the Hamming graphs by switching.

Godsil–McKay switching and twisted Grassmann graphs

We show that the twisted Grassmann graphs introduced by Van Dam and Koolen are obtained by Godsil-McKay switching applied to the Grassmann graphs. The partition for the switching is constructed by a polarity of a hyperplane.

Switched symplectic graphs and their 2-ranks

We apply Godsil–McKay switching to the symplectic graphs over $$\mathbb {F}_2$$ with at least 63 vertices and prove that the 2-rank of (the adjacency matrix of) the graph increases after switching. This shows that the switched graph is a new strongly regular graph with parameters $$(2^{2\nu }-1, 2^{2\nu -1}, 2^{2\nu -2},2^{2\nu -2})$$ and 2-rank $$2\nu +2$$ when $$\nu \ge 3$$ . For the symplectic graph on 63 vertices we investigate repeated switching by computer and find many new strongly regular graphs with the above parameters for $$\nu =3$$ with various 2-ranks. Using these results and a recursive construction method for the symplectic graph from Hadamard matrices, we obtain several graphs with the above parameters, but different 2-ranks for every $$\nu \ge 3$$ .

Godsil-McKay switching and isomorphism

Godsil-McKay switching is an operation on graphs that doesn’t change the spectrum of the adjacency matrix. Usually (but not always) the obtained graph is non-isomorphic with the original graph. We present a straightforward sufficient condition for being isomorphic after switching, and give examples which show that this condition is not necessary. For some graph products we obtain sufficient conditions for being non-isomorphic after switching. As an example we find that the tensor product of the grid L(ℓ,m) (ℓ > m>2) and a graph with at least one vertex of degree two is not determined by its adjacency spectrum.

E-cospectral hypergraphs and some hypergraphs determined by their spectra

Two k-uniform hypergraphs are said to be cospectral (E-cospectral), if their adjacency tensors have the same characteristic polynomial (E-characteristic polynomial). A k-uniform hypergraph H is said to be determined by its spectrum, if there is no non-isomorphic k-uniform hypergraph cospectral with H. In this note, we give a method for constructing E-cospectral hypergraphs, which is similar with Godsil–McKay switching. Some hypergraphs are shown to be determined by their spectra.

Cospectral Graphs and Regular Orthogonal Matrices of Level 2

For a graph $\Gamma$ with adjacency matrix $A$, we consider a switching operation that takes $\Gamma$ into a graph $\Gamma'$ with adjacency matrix $A'$, defined by $A'=Q^\top A Q$, where $Q$ is a regular orthogonal matrix of level $2$ (that is, $Q^\top Q=I$, $Q$1 $=$ 1, $2Q$ is integral, and $Q$ is not a permutation matrix). If such an operation exists, and $\Gamma$ is nonisomorphic with $\Gamma'$, then we say that $\Gamma'$ is semi-isomorphic with $\Gamma$. Semi-isomorphic graphs are $\mathbb {R}$-cospectral, which means that they are cospectral and so are their complements. Wang and Xu [On the asymptotic behavior of graphs determined by their generalized spectra, Discrete Math. 310 (2010)] expect that almost all pairs of nonisomorphic $\mathbb {R}$-cospectral graphs are semi-isomorphic.Regular orthogonal matrices of level $2$ have been classified. By use of this classification we work out the requirements for this switching operation to work in case $Q$ has one nontrivial indecomposable block of size $4$, $6$, $7$ or $8$. Size $4$ corresponds to Godsil-McKay switching. The other cases provide new methods for constructions of $\mathbb {R}$-cospectral graphs. For graphs with eight vertices all these constructions are carried out. As a result we find that, out of the 1166 graphs on eight vertices which are $\mathbb {R}$-cospectral to another graph, only 44 are not semi-isomorphic to another graph.

Cospectral Graphs and Regular Orthogonal Matrices of Level 2

For a graph Γ with adjacency matrix A, we consider a switching operation that takes Γ into a graph Γ′ with adjacency matrix A′, defined by A′ = Q^Τ AQ, where Q is a regular orthogonal matrix of level 2 (that is, Q^Τ Q = I, Q1 = 1, 2Q is integral, and Q is not a permutation matrix). If such an operation exists, and Γ is nonisomorphic with Γ′, then we say that Γ′ is semi-isomorphic with Γ. Semiisomorphic graphs are ℜ-cospectral, which means that they are cospectral and so are their complements. Wang and Xu [‘On the asymptotic behavior of graphs determined by their generalized spectra’, Discrete Math. 310 (2010)] expect that almost all pairs of ℜ-cospectral graphs are semi-isomorphic. Regular orthogonal matrices of level 2 have been classified. By use of this classification we work out the requirements for this switching operation to work in case Q has one nontrivial indecomposable block of size 4, 6, 7 or 8. Size 4 corresponds to Godsil-McKay switching. The other cases provide new methods for constructions of ℜ-cospectral graphs. For graphs with eight vertices all these constructions are carried out. As a result we find that, out of the 1166 graphs on eight vertices which are ℜ-cospectral to another graph, only 44 are not semi-isomorphic to another graph.