Det-extremal cubic bipartite graphs

Let G be a connected k–regular bipartite graph with bipartition V(G) = X∪Y and adjacency matrixA. We sayG is det–extremal ifper(A) = |det(A)|. Det–extremal k–regular bipartite graphs exist only for k = 2 or 3. McCuaig has characterized the det–extremal 3–connected cubic bipartite graphs. We extend McCuaig’s result by determining the structure of det–extremal cubic bipartite graphs of connectivity two. We use our results to determine which numbers can occur as orders of det-extremal connected cubic bipartite graphs, thus solving a problem due to H. Gropp.

[1]  William McCuaig,et al.  Pólya's Permanent Problem , 2004, Electron. J. Comb..

[2]  Harald Gropp.  Blocking set free configurations and their relations to digraphs and hypergraphs , 1997, Discret. Math..

[3]  C. Thomassen.  Sign-Nonsingular Matrices and Even Cycles in Directed Graphs , 1986 .

[4]  Branko Grünbaum,et al.  Are These Figures Oxymora , 1992 .

[5]  J. Edmonds,et al.  A Combinatorial Decomposition Theory , 1980, Canadian Journal of Mathematics.

[6]  F. Rendl,et al.  Operations preserving the pfaffian property of a graph , 1991, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[7]  P. W. Kasteleyn.  Dimer Statistics and Phase Transitions , 1963 .

[8]  Robin Thomas,et al.  PERMANENTS, PFAFFIAN ORIENTATIONS, AND EVEN DIRECTED CIRCUITS , 1997, STOC 1997.

[9]  C. Little.  A characterization of convertible (0, 1)-matrices , 1975 .

[10]  William McCuaig.  Even dicycles , 2000, J. Graph Theory.