We introduce the concept of a clique bitrade, which generalizes several known types of bitrades, including latin bitrades, Steiner T(k−1, k, v) bitrades, extended 1perfect bitrades. For a distance-regular graph, we show a one-to-one correspondence between the clique bitrades that meet the weight-distribution lower bound on the cardinality and the bipartite isometric subgraphs that are distance-...

In this paper we give an explicit formula for the least distortion embedding of a distance transitive graph into Euclidean space. We use this formula for finding least distortion embeddings for important examples: Hamming graphs, Johnson graphs, and Grassmann graphs. Our technique involves semidefinite programming and exploiting symmetry to simplify the optimization problem so that the question...

For a planar directed graph G, Postnikov’s boundary measurement map sends positive weight functions on the edges of G onto the appropriate totally nonnegative Grassmann cell. Assuming, without much loss of generality, that G is perfectly oriented, we establish an explicit formula for Postnikov’s map by expressing each Plücker coordinate as a ratio of two combinatorially defined polynomials in t...

In this paper we introduce Symplectic Grassmann codes, in analogy to ordinary Grassmann codes and Orthogonal Grassmann codes, as projective codes defined by symplectic Grassmannians. Lagrangian–Grassmannian codes are a special class of Symplectic Grassmann codes. We describe all the parameters of line Symplectic Grassmann codes and we provide the full weight enumerator for the Lagrangian–Grassm...

For a planar directed graph G, Postnikov’s boundary measurement map sends positive weight functions on the edges of G onto the appropriate totally nonnegative Grassmann cell. We establish an explicit formula for Postnikov’s map by expressing each Plücker coordinate as a ratio of two combinatorially defined polynomials in the edge weights, with positive integer coefficients. In the nonplanar set...

Let G be a perfectly oriented planar graph. Postnikov’s boundary measurement construction provides a rational map from the set of positive weight functions on the edges of G onto the appropriate totally nonnegative Grassmann cell. We establish an explicit combinatorial formula for Postnikov’s map by expressing each Plücker coordinate of the image as a ratio of two polynomials in the edge weight...

The present study introduces and investigates a new type of equation which is called Grassmann integral equation in analogy to integral equations studied in real analysis. A Grassmann integral equation is an equation which involves Grassmann (Berezin) integrations and which is to be obeyed by an unknown function over a (finite-dimensional) Grassmann algebra Gm (i.e., a sought after element of t...

We generalize the computation of Feynman integrals of log divergent graphs in terms of the Kirchhoff polynomial to the case of graphs with both fermionic and bosonic edges, to which we assign a set of ordinary and Grassmann variables. This procedure gives a computation of the Feynman integrals in terms of a period on a supermanifold, for graphs admitting a basis of the first homology satisfying...

Spatial search occurs in a connected graph if continuous-time quantum walk on the adjacency matrix of graph, suitably scaled, plus rank-one perturbation induced by any vertex will unitarily map principal eigenvector to characteristic vector vertex. This phenomenon is natural analogue Grover search. The spatial said be optimal it with constant fidelity and time inversely proportional shadow targ...

Learning on Grassmann manifold has become popular in many computer vision tasks, with the strong capability to extract discriminative information for imagesets and videos. However, such learning algorithms particularly on high-dimensional Grassmann manifold always involve with significantly high computational cost, which seriously limits the applicability of learning on Grassmann manifold in mo...