Given a graph G with n vertices, let p(G, j) denote the number of ways j mutually nonincident edges can be selected in G. The polynomial M(x) =∑[n/2] j=0 (−1) j p(G, j)xn−2 j , called the matching polynomial of G, is closely related to the Hosoya index introduced in applications in physics and chemistry. In this work we generalize this polynomial by introducing the number of disjoint paths of l...

We establish closed-form expansions for the universal edge elimination polynomial of paths and cycles and their generating functions. This includes closed-form expansions for the bivariate matching polynomial, the bivariate chromatic polynomial, and the covered components polynomial.

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. ...

In this paper, we study the signless Laplacian spectral radius of unicyclic graphs with prescribed number of pendant vertices or independence number. We also characterize the extremal graphs completely.

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...