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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
*L*_{T}
and *q*-Laplacian **L**_{T}^{q}.
Let GTS_{n} 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 *L*_{T} and
**L**_{T}^{q} as one moves up the poset
GTS_{n}.
Using the Frobenius characteristic, this can be thought as a result involving the
Schur symmetric function *s*_{λ}. 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
**L**_{T}^{q 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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