By Claude Berge (auth.), Siddani Bhaskara Rao (eds.)
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Additional info for Combinatorics and Graph Theory: Proceedings of the Symposium Held at the Indian Statistical Institute, Calcutta, February 25–29, 1980
THE PERMANENT POLYNOMIAL One reason why the characteristic polynomial failed to be a complete invariant may be because the determinant is not the appropriate matrix function relevant for combinatorial studies as by its very definition (with positive and negative terms) many terms get cancelled out, thereby losing significant combinatorial information. This observation leads us to consider the permanent as possibly the more appropriate matrix function for combinatorial studies. For any square matrix M of order n the perma- nent is defined by per(M) = (i) .
Matrix R In fact the can also be interpreted as though it represents a directed graph with feed- back loops between the nodes tether than a non-directed graph. For our requirements, Mason's theorem just gives the value of the determinant of R in terms of the products of the labels of edges in the loops, when Theorem 2. Determinant rii = 1. + (-1)m(sum of appropriately signed products of non-touching Feed-back loops taken Here the number m m at a time). is the maximum number of non-touching feed-back loops.
RP ... (28) indicates repeated application of the relevant operator (and not binomial expansion and substitution). Corollary 4. Since P(K I) = x we have n-1 = (x + 12V) n-1 P(K n) = 62 K 1 x, ... p(~n ) = 61n-i K 1 : (x + p2v)n-i x, ... (29) (30) and P(KI, n) = P(62K n) = (x +I2V)(x + p2v)n-lx . . (31) 6. A MULTIPLICATIVE POLYNOMIAL While the permanent polynomial P(G;x,I) and its homogenized version P(G;x,l,p) are conceptually the most natural ones they lack the very desirable property of being multiplicative over components of polynomial having the S r,s G.
Combinatorics and Graph Theory: Proceedings of the Symposium Held at the Indian Statistical Institute, Calcutta, February 25–29, 1980 by Claude Berge (auth.), Siddani Bhaskara Rao (eds.)