Laplacian Immanantal Polynomials of a Bipartite Graph and Graph Shift Operation

Let $G$ be a bipartite graph on $n$ vertices with the Laplacian matrix $L_G$. When $G$ is a tree, inequalities involving coefficients of immanantal polynomials of $L_G$ are known as we go up $GTS_n$ poset of unlabelled trees with $n$ vertices. We extend $GTS$ operation on a tree to an arbitrary graph, we call it generalized graph shift (hencefourth $GGS$) operation. Using $GGS$ operation, we generalize these known inequalities associated with trees to bipartite graphs. Using vertex orientations of $G$, we give a combinatorial interpretation for each coefficient of the Laplacian immanantal polynomial of $G$ which is used to prove counter parts of Schur theorem and Lieb's conjecture for these coefficients. We define $GGS_n$ poset on $\Omega_{C_k}^v(n)$, the set of unlabelled unicyclic graphs with $n$ vertices where each vertex of the cycle $C_k$ has degree $2$ except one vertex $v$. Using $GGS_n$ poset on $\Omega_{C_{2k}}^v(n)$, we solves an extreme value problem of finding the max-min pair in $\Omega_{C_{2k}}^v(n)$ for each coefficient of the generalized Laplacian polynomials. At the end of this paper, we also discuss the monotonicity of the spectral radius and the Wiener index of an unicyclic graph when we go up along $GGS_n$ poset of $\Omega_{C_k}^v(n)$.