---
title: Laplacian Permanental Polynomial
url: https://www.emergentmind.com/topics/laplacian-permanental-polynomial
type: topic
---

# Laplacian Permanental Polynomial

The **Laplacian permanental polynomial** of a graph \(G\) is the permanent-based analogue of the Laplacian characteristic polynomial, defined by
\[
\pi(L(G);x)=\operatorname{per}(xI-L(G)),
\]
where \(L(G)=D(G)-A(G)\) is the Laplacian matrix of \(G\), \(A(G)\) is the adjacency matrix, and \(D(G)\) is the degree matrix [2204.07798]. In the broader matrix-theoretic formulation, the permanent polynomial of a square matrix \(A\) is written as \(\psi(A)=\operatorname{per}(A-\lambda I)\), so a Laplacian matrix is a special case of the same construction [1701.04420]. The recent reconstruction literature also places \(\operatorname{per}(xI-L(G))\) inside the generalized family \(T_2(G;x)=\operatorname{per}(xI_n-\beta D-\gamma A)\), although that work does not isolate Laplacian permanental polynomials as a separate named object [2310.07104].

## 1. Definition, notation, and basic variants

For an \(n\times n\) matrix \(X=(x_{ij})\), the permanent is
\[
\operatorname{per}(X)=\sum_{\sigma}\prod_{i=1}^n x_{i,\sigma(i)},
\]
where the sum runs over all permutations \(\sigma\) of \(\{1,\dots,n\}\) [2204.07798]. The determinant differs in that it includes the sign of the permutation,
\[
\det(X)=\sum_{\sigma}\operatorname{sgn}(\sigma)\prod_{i=1}^n x_{i,\sigma(i)},
\]
whereas the permanent has no alternating sign [2204.07798]. The same paper recalls Valiant’s result that computing the permanent is \(\#P\)-complete even for \((0,1)\)-matrices [2204.07798].

For a graph \(G\) on vertex set \(V(G)=\{v_1,\dots,v_n\}\), with
\[
D(G)=\operatorname{diag}(d(v_1),d(v_2),\dots,d(v_n)),
\]
the standard matrices are
\[
L(G)=D(G)-A(G), \qquad Q(G)=D(G)+A(G),
\]
where \(Q(G)\) is the signless Laplacian matrix [2204.07798]. The two permanent-based graph polynomials most closely associated with Laplacian theory are
\[
\pi(L(G);x)=\operatorname{per}(xI-L(G)), \qquad
\pi(Q(G);x)=\operatorname{per}(xI-Q(G)) .
\]
The second is the **signless Laplacian permanental polynomial** [2204.07798].

Two graphs \(G\) and \(H\) are called **Laplacian copermanental** if
\[
\pi(L(G);x)=\pi(L(H);x),
\]
and **signless Laplacian copermanental** if
\[
\pi(Q(G);x)=\pi(Q(H);x) .
\]
A graph is said to be determined by one of these polynomials if equality of the corresponding polynomial forces isomorphism [2204.07798]. This shifts the central question from spectral cospectrality to permanent-based graph identification.

## 2. Matrix and digraph formulations

The matrix paper [1701.04420] represents every square matrix \(A=(a_{uv})\in\mathbb C^{n\times n}\) by a weighted digraph \(G(A)\): if \(a_{uv}\neq 0\), then \((u,v)\) is an arc of weight \(a_{uv}\), and a diagonal entry \(a_{uu}\) becomes a loop at \(u\) of weight \(a_{uu}\). For a Laplacian matrix \(L=D-A\), this means:

- diagonal entries \(L_{uu}=d(u)\) become loops of weight \(d(u)\),
- off-diagonal entries \(L_{uv}=-1\) become directed arcs of negative weight whenever vertices are adjacent,
- in the undirected case, the associated digraph is symmetric [1701.04420].

Within this framework, the Laplacian permanental polynomial is simply
\[
\psi(L)=\operatorname{per}(L-\lambda I) .
\]
The paper studies cut-vertices, blocks, pendant blocks, and induced subdigraph deletion, and uses them to derive recursive decompositions for permanent polynomials [1701.04420]. If \(G\) has a pendant block \(B_1\) attached at a cut-vertex \(v\) with loop weight \(\alpha\), then the basic recurrence is
\[
\psi(G)=\psi(B_1)\psi(G\setminus B_1)+\psi(B_1\setminus v)\psi(G\setminus(B_1\setminus v))
+(\lambda-\alpha)\psi(B_1\setminus v)\psi(G\setminus B_1).
\]
For Laplacians, \(\alpha=d(v)\), so the coefficient becomes \(\lambda-d(v)\) [1701.04420].

The same paper organizes these decompositions by \(\mathcal B\)-partitions. If \(G_k\) has blocks \(B_1,\dots,B_k\), a \(\mathcal B\)-partition is a partition into \(k\) vertex-disjoint induced subdigraphs \(\hat B_1,\dots,\hat B_k\) with \(\hat B_i\subseteq B_i\), and the associated \(\psi\)-summand is
\[
\prod_i \psi(\hat B_i),
\]
with \(\psi(\hat B_i)=1\) for a null graph [1701.04420]. The main structural formula expresses \(\psi(G_k)\) as a sum over deleted cut-vertex sets and over all \(\mathcal B\)-partition summands of the resulting induced subdigraphs. For Laplacian matrices, the coefficients contributed by deleted cut-vertices involve factors
\[
\prod_i (\lambda-d(v_i))(d_i-1),
\]
where \(d_i\) is the cut-index [1701.04420].

A key limitation is also stated explicitly: Schur complement methods are not available for permanents. The paper remarks that “The Schur's complement method is not applicable for finding out permanent of a matrix. It makes this problem complicated” [1701.04420]. For 2-connected graphs, therefore, no comparable permanent-specific simplification is supplied there.

## 3. Generalized permanental identities and reconstruction

The reconstruction paper [2310.07104] defines the generalized permanental polynomial
\[
T_2(G;x)=\operatorname{per}(xI_n-\beta D-\gamma A),
\]
where \(\beta,\gamma\in\mathbb R\) and \(\gamma\neq 0\). It explicitly defines the adjacency permanental polynomial
\[
o_4(G;x)=\operatorname{per}(xI-A),
\]
but it does **not** explicitly define a Laplacian permanental polynomial such as
\[
\operatorname{per}(xI-L(G))=\operatorname{per}(xI-D+A)
\]
or a signless-Laplacian permanental polynomial
\[
\operatorname{per}(xI-Q(G))=\operatorname{per}(xI-D-A).
\]
Instead, both are naturally contained in \(T_2(G;x)\): the Laplacian case corresponds to \((\beta,\gamma)=(1,-1)\), and the signless Laplacian case to \((\beta,\gamma)=(1,1)\) [2310.07104].

The main identity for \(T_2\) is
\[
(m-n)T_2(G;x)+xT_2'(G;x)
=
\sum_{e\in E(G)}T_2(G-e;x)
-(\gamma^2+\beta^2)\sum_{v_sv_t\in E(G)}
\operatorname{per}\!\left[(xI_n-\beta D-\gamma A)^{s,t}_{s,t}\right].
\]
Substituting \((\beta,\gamma)=(1,-1)\) yields the natural Laplacian permanental relation
\[
(m-n)\operatorname{per}(xI-D+A)+x\frac{d}{dx}\operatorname{per}(xI-D+A)
=
\sum_{e\in E(G)}\operatorname{per}(xI-D_e+A_e)
-2\sum_{v_sv_t\in E(G)}
\operatorname{per}\!\left[(xI_n-D+A)^{s,t}_{s,t}\right].
\]
The corresponding signless-Laplacian formula has the same coefficient \(-2\) [2310.07104].

The contrast with determinant-based Laplacian theory is structural. In the generalized determinant identity, the correction term is multiplied by \(\gamma^2-\beta^2\), so it vanishes for \((\beta,\gamma)=(1,\pm1)\); this is why the paper proves especially clean edge-reconstruction theorems for the Laplacian and signless Laplacian **characteristic** polynomials [2310.07104]. In the permanent case, however, the correction coefficient is \(-(\gamma^2+\beta^2)\), which becomes \(-2\) at \((1,\pm1)\). This strongly suggests that a Laplacian permanental reconstruction theory is substantially less direct. The paper does not isolate these objects as separate graph polynomials and does not prove dedicated reconstruction theorems for them [2310.07104].

## 4. Coefficient formulas, recurrences, and combinatorial structure

For
\[
\pi(L(G);x)=\sum_i p_i(G)x^{n-i},
\]
the bicyclic-graph paper gives the low-order coefficients
\[
p_0(G)=1,\qquad
p_1(G)=-2m,\qquad
p_2(G)=2m^2+m-\frac12\sum_i d_i^2,
\]
and
\[
p_3(G)= -\frac13\sum_i d_i^3 +(m+1)\sum_i d_i^2 -\frac43 m^3-2m^2+2c_3(G).
\]
It also supplies an explicit formula for \(p_4(G)\) involving \(\sum_i d_i^4\), \(\sum_{(v_iv_j)\in E(G)}d_id_j\), \(c_3(G)\), \(c_4(G)\), and \(\sum_i d_i c_3(G_{v_i})\) [2204.07798]. For the signless Laplacian permanental polynomial
\[
\pi(Q(G);x)=\sum_i q_i(G)x^{n-i},
\]
the formulas for \(q_0,q_1,q_2\) are the same, while the cubic coefficient becomes
\[
q_3(G)= -\frac13\sum_i d_i^3 +(m+1)\sum_i d_i^2 -\frac43 m^3-2m^2-2c_3(G),
\]
and an explicit \(q_4(G)\) is also given [2204.07798].

These formulas imply that \(\pi(L(G);x)\) determines:

1. the number of vertices,
2. the number of edges,
3. the sum of squares of degrees,
4. the quantity \(\sum_i d_i^3-6c_3(G)\),

while \(\pi(Q(G);x)\) determines the analogous quantity \(\sum_i d_i^3+6c_3(G)\) [2204.07798]. This coefficient extraction is one of the main routes by which permanental polynomials are turned into graph-characterization tools.

The same paper uses vertex-deletion and cycle-expansion recurrences. If \(v\) is a vertex with neighborhood \(N(v)\) and \(\mathscr C_G(v)\) is the set of cycles containing \(v\), then
\[
\pi(L(G);x) = (x-d(v))\pi(L_v(G)) +\sum_{u\in N(v)}\pi(L_{uv}(G)) +2\sum_{C\in \mathscr C_G(v)} \pi(L_{V(C)}(G)),
\]
and
\[
\pi(Q(G);x) = (x-d(v))\pi(Q_v(G)) +\sum_{u\in N(v)}\pi(Q_{uv}(G)) +2\sum_{C\in \mathscr C_G(v)}(-1)^{|V(C)|}\pi(Q_{V(C)}(G)).
\]
For coalescence \(G\cdot H\), obtained by identifying \(u\in V(G)\) with \(v\in V(H)\), the paper uses
\[
\pi(L(G\cdot H)) = \pi(L(G))\pi(L_v(H)) +\pi(L(H))\pi(L_u(G)) -x\,\pi(L_v(H))\pi(L_u(G)),
\]
with the same formal pattern in the signless case [2204.07798].

A distinct combinatorial formula is given for the permanent of the Laplacian matrix itself. If \(H(K)\) is the set of subgraphs on vertex set \(\{v_i:i\in K\}\) such that each component is either an edge or a cycle, and \(c(H)\) is the number of cyclic components, then
\[
\operatorname{per}(L(G)) =
\sum_{K\subseteq \{1,\dots,n\}}
\sum_{H\in H(K)}
(-1)^{|K|}2^{c(H)}\prod_{i\notin K} d_i .
\]
The signless analogue is
\[
\operatorname{per}(Q(G)) =
\sum_{K\subseteq \{1,\dots,n\}}
\sum_{H\in H(K)}
2^{c(H)}\prod_{i\notin K} d_i .
\]
These formulas are used in uniqueness arguments inside structured graph families [2204.07798].

## 5. Bicyclic graph families and copermanental characterization

A bicyclic graph is a connected graph with exactly two independent cycles; equivalently, a connected graph on \(n\) vertices with \(n+1\) edges [2204.07798]. The paper [2204.07798] studies two standard bicyclic families.

The graph \(d(p,q,r)\) is obtained by identifying the cycles \(C_p\) and \(C_q\) with two different end vertices of the path \(P_r\), where \(p,q\ge 3\). Its order is
\[
n=p+q+r.
\]
The graph \(\theta(p,q,r)\) consists of two fixed vertices joined by three internally disjoint paths of orders \(p,q,r\), where \(p,q,r\ge 0\) and at most one of them is \(0\). Its order is
\[
n=p+q+r+2.
\]
The principal theorems are:

- \(d(p,q,r)\) is determined by its Laplacian permanental polynomial;
- \(\theta(p,q,r)\) is determined by its Laplacian permanental polynomial;
- \(d(p,q,r)\) is determined by its signless Laplacian permanental polynomial;
- \(\theta(p,q,r)\) is determined by its signless Laplacian permanental polynomial [2204.07798].

The proofs combine coefficient identities, recurrence formulas, coalescence formulas, transformed explicit expressions, and special evaluations at \(y=1\). A central structural fact is that both families are nearly regular with degree sequence
\[
(3^2,2^{n-2}),
\]
and the paper uses the lemma that if \(G\) is nearly regular and
\[
\pi(L(H);x)=\pi(L(G);x)
\quad\text{or}\quad
\pi(Q(H);x)=\pi(Q(G);x),
\]
then \(H\) has the same degree sequence as \(G\) [2204.07798]. This sharply restricts the possible copermanental mates.

The internal uniqueness step compares transformed expressions \(f_L(p,q,r;y)\) or \(f_Q(p,q,r;y)\). The largest exponents in those expressions depend linearly on \(p,q,r\), so equality of polynomials forces equality of the exponent patterns, which in turn determines \(p,q,r\), up to the symmetry \(p\leftrightarrow q\) in \(d(p,q,r)\) [2204.07798]. Disconnected competitors are then ruled out by the evaluations
\[
\pi(L(C_n);1)=(-1)^n+3,\qquad
\pi(Q(C_n);1)=3(-1)^n+1,
\]
together with explicit \(y=1\) values for the target families [2204.07798].

The same paper also notes a relation of Faria: for a bipartite graph \(G\),
\[
\pi(Q(G);x)=\pi(L(G);x).
\]
This does not apply to all bicyclic graphs, but it places the Laplacian and signless Laplacian permanental invariants in especially close alignment on bipartite subclasses [2204.07798].

## 6. Structured families, later developments, and open problems

The 2025 paper [2509.14389] studies several further families: coconut trees \(_{m,n}\), regular spider trees \(\mathcal S_{n,m}\), perfect binary trees \(T_\ell\), and corona products such as \(C_n\odot K_1\), \(C_3\odot \bar K_n\), and \(C_4\odot \bar K_n\). It defines the same two graph polynomials
\[
\psi(L(G);x)=\operatorname{per}(xI_n-L(G)), \qquad
\psi(Q(G);x)=\operatorname{per}(xI_n-Q(G)),
\]
and frames the central question as whether a graph is determined by one polynomial, by the other, or only by the pair [2509.14389].

For bipartite graphs, the paper repeatedly uses
\[
\psi(L(G);x)=\psi(Q(G);x),
\]
so one calculation yields both polynomials [2509.14389]. This simplification underlies the treatment of coconut trees, spider trees, and perfect binary trees.

Several exact polynomial recurrences are obtained. For the regular spider \(\mathcal S_{n,m}\),
\[
\psi(L(\mathcal S_{n,m}); x) = \psi(Q(\mathcal S_{n,m}); x) = (x-n)C_m^n + nC_{m-1}C_m^{n-1},
\]
where
\[
C_1=x-1,\qquad C_2=(x-2)(x-1)+1,\qquad C_k=(x-2)C_{k-1}+C_{k-2}.
\]
For the perfect binary tree \(T_\ell\),
\[
\psi(L(T_{\ell}); x) = \psi(Q(T_\ell); x) = (x-2)A_\ell^2 + 2 A_{\ell} A_{\ell-1}^{2},
\]
with
\[
A_0 = 1,\qquad A_1 = x-1,\qquad A_i = (x-3) A_{i-1}^2 + 2 A_{i-1}A_{i-2}^2 .
\]
These formulas provide general computation schemes, but the characterization theorems proved in the paper are more limited [2509.14389].

The paper proves that the following graphs are determined by their Laplacian permanental polynomials, and also by their signless Laplacian permanental polynomials in the cases stated:

- the coconut trees \(_{3,n}\) and \(_{4,n}\),
- the spider \(S_{n,2}\),
- the perfect binary trees \(T_2\) and \(T_3\),
- \(C_3\odot K_1\), \(C_4\odot K_1\), and \(C_5\odot K_1\),
- \(C_3\odot \bar K_n\),
- \(C_4\odot \bar K_n\) [2509.14389].

At the same time, the scope limitations are explicit. The case \(S_{n,m}\) for \(m\ge 3\) remains open; the paper gives a recursive formula for all perfect binary trees \(T_\ell\), but proves characterization only for \(T_2\) and \(T_3\); and it states the conjecture that for \(n\ge 6\), \(C_n\odot K_1\) is determined by its (signless) Laplacian permanental polynomials [2509.14389]. The paper also remarks that the body of the manuscript does not present a concrete family in which neither polynomial alone determines the graph but the pair of both does [2509.14389].

A further textual issue is documented in the synthesis: some displayed formulas in the manuscript contain typographical inconsistencies, especially in the \(C_5\odot K_1\), \(C_3\odot \bar K_n\), and \(C_4\odot \bar K_n\) sections [2509.14389]. This does not alter the paper’s stated characterization theorems, but it affects the reliability of some printed intermediate expressions.

Across these works, a coherent picture emerges. The Laplacian permanental polynomial is explicitly defined and used as a graph-characterizing invariant for several structured families [2204.07798, 2509.14389]; it admits recursive and block-decomposition methods through weighted digraphs, cut-vertices, and \(\mathcal B\)-partitions [1701.04420]; and it fits naturally into the generalized permanent framework
\[
\operatorname{per}(xI_n-\beta D-\gamma A),
\]
where Laplacian and signless Laplacian specializations are immediate but technically more resistant to edge-reconstruction arguments than the determinant-based Laplacian characteristic polynomial [2310.07104]. This suggests that the subject sits at the intersection of permanental combinatorics, graph reconstruction, and graph characterization by polynomial invariants.

Source: https://www.emergentmind.com/topics/laplacian-permanental-polynomial