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Let $G$ be a bipartite graph on $n$ vertices with the Laplacian matrix $L_G$.When $G$ is a tree, inequalities involving coefficients of immanantalpolynomials of $L_G$ are known as we go up $GTS_n$ poset of unlabelled treeswith $n$ 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 tobipartite graphs. Using vertex orientations of $G$, we give a combinatorialinterpretation for each coefficient of the Laplacian immanantal polynomial of$G$ which is used to prove counter parts of Schur theorem and Lieb's conjecturefor these coefficients. We define $GGS_n$ poset on $\Omega_{C_k}^v(n)$, the setof 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 themax-min pair in $\Omega_{C_{2k}}^v(n)$ for each coefficient of the generalizedLaplacian polynomials. At the end of this paper, we also discuss themonotonicity of the spectral radius and the Wiener index of an unicyclic graphwhen we go up along $GGS_n$ poset of $\Omega_{C_k}^v(n)$.

Laplacian Immanantal Polynomials of a Bipartite Graph and Graph Shift Operation