In this paper, we study graphs with integer matching polynomial roots. We characterize all traceable graphs whose all matching polynomial roots are integer. We show that there is no k-regular claw-free graph (k ≥ 2), whose all matching polynomial roots are integer. Also we conjecture that apart from K7 \ (E(C3) ∪E(C4)), there is no matching integral k-regular graph (k ≥ 2). 2010 Mathematics Sub...

We discuss the complexity of computing various graph polynomials of graphs of fixed clique-width. We show that the chromatic polynomial, the matching polynomial and the two-variable interlace polynomial of a graph G of clique-width at most k with n vertices can be computed in time O(n), where f(k) ≤ 3 for the inerlace polynomial, f(k) ≤ 2k + 1 for the matching polynomial and f(k) ≤ 3 · 2 for th...

Let the matching polynomial of a graph G be denoted by μ(G, x). A graph G is said to be θ-super positive if μ(G, θ) 6= 0 and μ(G \ v, θ) = 0 for all v ∈ V (G). In particular, G is 0-super positive if and only if G has a perfect matching. While much is known about 0-super positive graphs, almost nothing is known about θsuper positive graphs for θ 6= 0. This motivates us to investigate the struct...

Polyomino graphs is one of the research objectives in statistical physics and in modeling problems of surface chemistry. A random polyomino chain is a subgraph of a polyomino graph. The matching energy is defined as the sum of the absolute values of the zeros of the matching polynomial of a graph. In this paper, we characterize the graphs with the extremal matching energy among all random polyo...

Let H = (V, E) be a 3-uniform linear hypergraph with one hypercycle C3. We consider a blow-up hypergraph B[H]. We are interested in the following problem. We have to decide whether there exists a blow-up hypergraph B[H] of the hypergraph H, with hyperedge densities satisfying special conditions, such that the hypergraph H appears in a blow-up hypergraph as a transversal. We present an efficient...

Godsil observed the simple fact that the multiplicity of 0 as a root of the matching polynomial of a graph coincides with the classical notion of deficiency. From this fact he asked to what extent classical results in matching theory generalize, replacing “deficiency” with multiplicity of θ as a root of the matching polynomial. We prove an analogue of the Stability Lemma for any given root, whi...