  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 $\Omega_{C_k}^v(n)$.
