Edge-local equivalence of graphs

The local complement G*i of a simple graph G at one of its vertices i is obtained by complementing the subgraph induced by the neighborhood of i and leaving the rest of the graph unchanged. If e={i,j} is an edge of G then G*e=((G*i)*j)*i is called the edge-local complement of G along the edge e. We call two graphs edge-locally equivalent if they are related by a sequence of edge-local complementations. The main result of this paper is an algebraic description of edge-local equivalence of graphs in terms of linear fractional transformations of adjacency matrices. Applications of this result include (i) a polynomial algorithm to recognize whether two graphs are edge-locally equivalent, (ii) a formula to count the number of graphs in a class of edge-local equivalence, and (iii) a result concerning the coefficients of the interlace polynomial, where we show that these coefficients are all even for a class of graphs; this class contains, as a subset, all strongly regular graphs with parameters (n, k, a, c), where k is odd and a and c are even.

[1]  B. De Moor,et al.  Finite set of invariants to characterize local Clifford equivalence of stabilizer states , 2005 .

[2]  A. Bouchet.  Digraph decompositions and Eulerian systems , 1987 .

[3]  Matthew G. Parker,et al.  Generalized Bent Criteria for Boolean Functions (I) , 2005, IEEE Transactions on Information Theory.

[4]  André Bouchet,et al.  Isotropic Systems , 1987, Eur. J. Comb..

[5]  H. Briegel,et al.  Measurement-based quantum computation on cluster states , 2003, quant-ph/0301052.

[6]  Béla Bollobás,et al.  The interlace polynomial: a new graph polynomial , 2000, SODA '00.

[7]  J. Eisert,et al.  Multiparty entanglement in graph states , 2003, quant-ph/0307130.

[8]  Béla Bollobás,et al.  Alternating Knot Diagrams, Euler Circuits and the Interlace Polynomial , 2001, Eur. J. Comb..

[9]  André Bouchet,et al.  Recognizing locally equivalent graphs , 1993, Discret. Math..

[10]  Daniel Gottesman,et al.  Stabilizer Codes and Quantum Error Correction , 1997, quant-ph/9705052.

[11]  André Bouchet,et al.  Graphic presentations of isotropic systems , 1987, J. Comb. Theory, Ser. B.

[12]  Maarten Van den Nest,et al.  Local equivalence of stabilizer states and codes , 2005 .

[13]  André Bouchet,et al.  Reducing prime graphs and recognizing circle graphs , 1987, Comb..

[14]  André Bouchet.  Transforming trees by successive local complementations , 1988, J. Graph Theory.

[15]  Béla Bollobás,et al.  The Interlace Polynomial of Graphs at - 1 , 2002, Eur. J. Comb..

[16]  A. Bouchet.  Connectivity of Isotropic Systems , 1989 .