A refined Gallai-Edmonds structure theorem for weighted matching polynomials

Abstract

In this work, we prove a refinement of the Gallai-Edmonds structure theorem for weighted matching polynomials by Ku and Wong. Our proof uses a connection between matching polynomials and branched continued fractions. We also show how this is related to a modification by Sylvester of the classical Sturm's theorem on the number of zeros of a real polynomial in an interval. In addition, we obtain some other results about zeros of matching polynomials.

Figures (5)

• Figure 1: A rooted tree and its associated tree continued fraction.
• Figure 2: An illustration of the equality $\alpha_i(G)=\alpha_i(T^i_G)$.
• Figure 3: An example of a graph of $\alpha_i(G)(x)$.
• Figure 4: A rooted tree with vertex weights $x$ and edge weights $-1$. The signs are of the graph continued fractions of the subtrees. As time passes the plus signs fill in the tree.
• Figure 5: The nodes represent possible signs for the pair of distinct vertices $(i,j)$, both in $G$ and in $G\setminus j$ and $G\setminus i$. The edges join signs configurations that can occur simultaneously. The green, black and yellow edges represent $\lambda_{i\sim j}$ in $[-\infty,0]$, $(-\infty,0]$ and $(-\infty,0)$, respectively. The red and blue edges represent $\lambda_{i\sim j}$ equal to $-\infty$ and $0$, respectively.

Theorems & Definitions (69)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Theorem 1
• Theorem F
• Theorem 2
• Lemma 3
• proof