Hafnian of two-parameter matrices

The concept of the hafnian first appeared in works on
 quantum field theory by E. R. Caianiello. However, it also has
 an important combinatorial property: adjacency
 matrix undirected weighted graph is equal to the
 total sum weights perfect matchings this graph.
 In general, use limited complexity
 its computation. paper, we present a method for exact calculation two-parameter matrices.
 terms graphs, count of
 graphs whose edge take only
 two values. This based on formula expressing
 matrices through product hafnians their submatrices. necessary condition
 application possibility count
 number k-edge some graphs. We consider
 special case detail using Toeplitz as matrix. As example, propose new interpretation
 sequences from On-Line Encyclopedia
 Integer Sequences and then provide analytical
 formulas certain linear chord
 diagrams.