Séminaire Lotharingien de Combinatoire, B83a (2021), 10 pp.

Mukesh Kumar Nagar and Sivaramakrishnan Sivasubramanian

Generalized Matrix Polynomials of Tree Laplacians Indexed by Symmetric Functions and the GTS Poset

Abstract. Let T be a tree on n vertices with Laplacian matrix LT and q-Laplacian LTq. Let GTSn be the generalized tree shift poset on the set of unlabelled trees on n vertices. Inequalities are known between coefficients of the immanantal polynomial of LT and LTq as one moves up the poset GTSn. Using the Frobenius characteristic, this can be thought as a result involving the Schur symmetric function sλ. In this paper, we use an arbitrary symmetric function to define a generalized matrix function of an n x n matrix. When the symmetric function is the monomial and the forgotten symmetric function, we generalize such inequalities among coefficients of the generalized matrix polynomial of LTq as one moves up the GTSn poset.

Received: December 6, 2019. Revised;: April 30, 2021. Accepted: May 6, 2021. Final Version: May 9, 2021.

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