Avoiding Root Coincidences in the Weighted Matching Polynomial

Johnna Parenteau (University of Regina)

Historically, the characteristic polynomial has primarily been the object used to drive spectral information about a graph; however, the weighted matching polynomial, denoted by \(m_w(G,x)\), has recently been a topic of interest due to its inherent connection to \(k\)-matchings and other graphical properties. It is well known the roots of the weighted matching polynomial enjoy an interlacing property with respect to a vertex deleted subgraph. A natural question is for which graphs \(G\) does strict interlacing hold among the roots of \(m_w(G,x)\) and \(m_w(G\setminus \{v\},x)\)? A classical example of such a graph is one that contains a Hamilton path. In this talk, we consider extending the class of such graphs, called SRSI graphs, and demonstrate how these graphs can be constructed inductively. Additionally, we determine the only trees that are SRSI for all weightings and with respect to all vertices are \(K_2\) and \(P_4\).

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