The independence polynomial of a graph G is the polynomial ∑ I x, summed over all independent subsets I ⊆ V (G). We show that if G is clawfree, then there exists a Mehler formula for its independence polynomial. This was proved for matching polynomials in [18] and extends the combinatorial proof of the Mehler formula imagined by Foata [9]. It implies immediately that all the roots of the indepe...

Here, we find the characteristics polynomial of normalized Laplacian of a tree. The coefficients of this polynomial are expressed by the higher order general Randić indices for matching, whose values depend on the structure of the tree. We also find the expression of these indices for starlike tree and a double-starlike tree, Hm(p, q). Moreover, we show that two cospectral Hm(p, q) of the same ...

IV-matching is a generalization of perfect bipartite matching. The complexity of finding IV-matching in a graph was posted as an open problem at the ICALP 2014 conference. In this note, we resolve the question and prove that, contrary to the expectations of the authors, the given problem is strongly NP-hard (already in the simplest non-trivial case of four layers). Hence it is unlikely that the...

Let k ≥ 2, m ≥ 5 and n = mk be integers. By finding bounds for certain rook polynomials, we identify the k×n Latin rectangles with the most extensions to (k+1)×n Latin rectangles. Equivalently, we find the (n− k)-regular subgraphs of Kn,n which have the greatest number of perfect matchings, and the (0, 1)-matrices with exactly k zeroes in every row and column which maximise the permanent. Witho...

How many perfect matchings are contained in a given bipartite graph? An exercise in Godsil’s 1993 Algebraic Combinatorics solicits proof that this question’s answer is an integral involving a certain rook polynomial. Though not widely known, this result appears implicitly in Riordan’s 1958 An Introduction to Combinatorial Analysis. It was stated more explicitly and proved independently by S.A. ...

A minimal blocker in a bipartite graph G is a minimal set of edges the removal of which leaves no perfect matching in G. We give an explicit characterization of the minimal blockers of a bipartite graph G. This result allows us to obtain a polynomial delay algorithm for finding all minimal blockers of a given bipartite graph. Equivalently, this gives a polynomial delay algorithm for listing the...

This paper addresses the choice of an optimal smoothing parameter for local polynomial matching. A version of Empirical Bias Bandwidth Selection (EBBS) proposed by Ruppert (1997) is applied to account for the MSE computation of the matching estimator. Thereby, an estimator for the large sample variance of the local polynomial matching estimator is also provided. A Monte Carlo study indicates be...

We study uncoordinated matching markets with additional local constraints that capture, e.g., restricted information, visibility, or externalities in markets. Each agent is a node in a fixed matching network and strives to be matched to another agent. Each agent has a complete preference list over all other agents it can be matched with. However, depending on the constraints and the current sta...

We consider the approximability of three recently introduced pattern matching problems which have been shown to be NP-hard. Given two strings as input, the first problem is to find the longest common parameterised subsequence between two strings. The second is a maximisation variant of generalised function matching and the third is a a maximisation variant of generalised parameterised matching....

We study various versions of the univariate and multivariate matching and rook polynomials. We show that there is most general multivariate matching polynomial, which is, up the some simple substitutions and multiplication with a prefactor, the original multivariate matching polynomial introduced by C. Heilmann and E. Lieb. We follow here a line of investigation which was very successfully purs...