Laplacian Permanental Polynomial

• Laplacian permanental polynomial is defined as per(xI - L(G)) and serves as a permanent-based analog to the Laplacian characteristic polynomial.
• Recursive decompositions and block-partition methods are employed to compute and analyze the polynomial, especially via cut-vertices and induced subdigraphs.
• Explicit coefficient formulas and recurrences facilitate graph characterization and help distinguish structured families like bicyclic and nearly regular graphs.

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 (Wu et al., 2022). 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 (Singh et al., 2017). 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 (Zhang et al., 2023).

1. Definition, notation, and basic variants

For an $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$0 matrix $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$1, the permanent is

$\pi(L(G);x)=\operatorname{per}(xI-L(G)),$2

where the sum runs over all permutations $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$3 of $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$4 (Wu et al., 2022). The determinant differs in that it includes the sign of the permutation,

$\pi(L(G);x)=\operatorname{per}(xI-L(G)),$5

whereas the permanent has no alternating sign (Wu et al., 2022). The same paper recalls Valiant’s result that computing the permanent is $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$6-complete even for $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$7-matrices (Wu et al., 2022).

For a graph $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$8 on vertex set $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$9, with

$L(G)=D(G)-A(G)$0

the standard matrices are

$L(G)=D(G)-A(G)$1

where $L(G)=D(G)-A(G)$2 is the signless Laplacian matrix (Wu et al., 2022). The two permanent-based graph polynomials most closely associated with Laplacian theory are

$L(G)=D(G)-A(G)$3

The second is the signless Laplacian permanental polynomial (Wu et al., 2022).

Two graphs $L(G)=D(G)-A(G)$4 and $L(G)=D(G)-A(G)$5 are called Laplacian copermanental if

$L(G)=D(G)-A(G)$6

and signless Laplacian copermanental if

$L(G)=D(G)-A(G)$7

A graph is said to be determined by one of these polynomials if equality of the corresponding polynomial forces isomorphism (Wu et al., 2022). This shifts the central question from spectral cospectrality to permanent-based graph identification.

2. Matrix and digraph formulations

The matrix paper (Singh et al., 2017) represents every square matrix $L(G)=D(G)-A(G)$8 by a weighted digraph $L(G)=D(G)-A(G)$9: if $G$0, then $G$1 is an arc of weight $G$2, and a diagonal entry $G$3 becomes a loop at $G$4 of weight $G$5. For a Laplacian matrix $G$6, this means:

• diagonal entries $G$7 become loops of weight $G$8,
• off-diagonal entries $G$9 become directed arcs of negative weight whenever vertices are adjacent,
• in the undirected case, the associated digraph is symmetric (Singh et al., 2017).

Within this framework, the Laplacian permanental polynomial is simply

$A(G)$0

The paper studies cut-vertices, blocks, pendant blocks, and induced subdigraph deletion, and uses them to derive recursive decompositions for permanent polynomials (Singh et al., 2017). If $A(G)$1 has a pendant block $A(G)$2 attached at a cut-vertex $A(G)$3 with loop weight $A(G)$4, then the basic recurrence is

$A(G)$5

For Laplacians, $A(G)$6, so the coefficient becomes $A(G)$7 (Singh et al., 2017).

The same paper organizes these decompositions by $A(G)$8-partitions. If $A(G)$9 has blocks $D(G)$0, a $D(G)$1-partition is a partition into $D(G)$2 vertex-disjoint induced subdigraphs $D(G)$3 with $D(G)$4, and the associated $D(G)$5-summand is

$D(G)$6

with $D(G)$7 for a null graph (Singh et al., 2017). The main structural formula expresses $D(G)$8 as a sum over deleted cut-vertex sets and over all $D(G)$9-partition summands of the resulting induced subdigraphs. For Laplacian matrices, the coefficients contributed by deleted cut-vertices involve factors

$A$0

where $A$1 is the cut-index (Singh et al., 2017).

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” (Singh et al., 2017). For 2-connected graphs, therefore, no comparable permanent-specific simplification is supplied there.

3. Generalized permanental identities and reconstruction

The reconstruction paper (Zhang et al., 2023) defines the generalized permanental polynomial

$A$2

where $A$3 and $A$4. It explicitly defines the adjacency permanental polynomial

$A$5

but it does not explicitly define a Laplacian permanental polynomial such as

$A$6

or a signless-Laplacian permanental polynomial

$A$7

Instead, both are naturally contained in $A$8: the Laplacian case corresponds to $A$9, and the signless Laplacian case to $\psi(A)=\operatorname{per}(A-\lambda I)$0 (Zhang et al., 2023).

The main identity for $\psi(A)=\operatorname{per}(A-\lambda I)$1 is

$\psi(A)=\operatorname{per}(A-\lambda I)$2

Substituting $\psi(A)=\operatorname{per}(A-\lambda I)$3 yields the natural Laplacian permanental relation

$\psi(A)=\operatorname{per}(A-\lambda I)$4

The corresponding signless-Laplacian formula has the same coefficient $\psi(A)=\operatorname{per}(A-\lambda I)$5 (Zhang et al., 2023).

The contrast with determinant-based Laplacian theory is structural. In the generalized determinant identity, the correction term is multiplied by $\psi(A)=\operatorname{per}(A-\lambda I)$6, so it vanishes for $\psi(A)=\operatorname{per}(A-\lambda I)$7; this is why the paper proves especially clean edge-reconstruction theorems for the Laplacian and signless Laplacian characteristic polynomials (Zhang et al., 2023). In the permanent case, however, the correction coefficient is $\psi(A)=\operatorname{per}(A-\lambda I)$8, which becomes $\psi(A)=\operatorname{per}(A-\lambda I)$9 at $\operatorname{per}(xI-L(G))$0. 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 (Zhang et al., 2023).

4. Coefficient formulas, recurrences, and combinatorial structure

For

$\operatorname{per}(xI-L(G))$1

the bicyclic-graph paper gives the low-order coefficients

$\operatorname{per}(xI-L(G))$2

and

$\operatorname{per}(xI-L(G))$3

It also supplies an explicit formula for $\operatorname{per}(xI-L(G))$4 involving $\operatorname{per}(xI-L(G))$5, $\operatorname{per}(xI-L(G))$6, $\operatorname{per}(xI-L(G))$7, $\operatorname{per}(xI-L(G))$8, and $\operatorname{per}(xI-L(G))$9 (Wu et al., 2022). For the signless Laplacian permanental polynomial

$T_2(G;x)=\operatorname{per}(xI_n-\beta D-\gamma A)$0

the formulas for $T_2(G;x)=\operatorname{per}(xI_n-\beta D-\gamma A)$1 are the same, while the cubic coefficient becomes

$T_2(G;x)=\operatorname{per}(xI_n-\beta D-\gamma A)$2

and an explicit $T_2(G;x)=\operatorname{per}(xI_n-\beta D-\gamma A)$3 is also given (Wu et al., 2022).

These formulas imply that $T_2(G;x)=\operatorname{per}(xI_n-\beta D-\gamma A)$4 determines:

1. the number of vertices,
2. the number of edges,
3. the sum of squares of degrees,
4. the quantity $T_2(G;x)=\operatorname{per}(xI_n-\beta D-\gamma A)$5,

while $T_2(G;x)=\operatorname{per}(xI_n-\beta D-\gamma A)$6 determines the analogous quantity $T_2(G;x)=\operatorname{per}(xI_n-\beta D-\gamma A)$7 (Wu et al., 2022). 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 $T_2(G;x)=\operatorname{per}(xI_n-\beta D-\gamma A)$8 is a vertex with neighborhood $T_2(G;x)=\operatorname{per}(xI_n-\beta D-\gamma A)$9 and $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$00 is the set of cycles containing $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$01, then

$\pi(L(G);x)=\operatorname{per}(xI-L(G)),$02

and

$\pi(L(G);x)=\operatorname{per}(xI-L(G)),$03

For coalescence $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$04, obtained by identifying $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$05 with $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$06, the paper uses

$\pi(L(G);x)=\operatorname{per}(xI-L(G)),$07

with the same formal pattern in the signless case (Wu et al., 2022).

A distinct combinatorial formula is given for the permanent of the Laplacian matrix itself. If $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$08 is the set of subgraphs on vertex set $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$09 such that each component is either an edge or a cycle, and $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$10 is the number of cyclic components, then

$\pi(L(G);x)=\operatorname{per}(xI-L(G)),$11

The signless analogue is

$\pi(L(G);x)=\operatorname{per}(xI-L(G)),$12

These formulas are used in uniqueness arguments inside structured graph families (Wu et al., 2022).

5. Bicyclic graph families and copermanental characterization

A bicyclic graph is a connected graph with exactly two independent cycles; equivalently, a connected graph on $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$13 vertices with $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$14 edges (Wu et al., 2022). The paper (Wu et al., 2022) studies two standard bicyclic families.

The graph $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$15 is obtained by identifying the cycles $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$16 and $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$17 with two different end vertices of the path $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$18, where $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$19. Its order is

$\pi(L(G);x)=\operatorname{per}(xI-L(G)),$20

The graph $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$21 consists of two fixed vertices joined by three internally disjoint paths of orders $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$22, where $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$23 and at most one of them is $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$24. Its order is

$\pi(L(G);x)=\operatorname{per}(xI-L(G)),$25

The principal theorems are:

• $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$26 is determined by its Laplacian permanental polynomial;
• $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$27 is determined by its Laplacian permanental polynomial;
• $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$28 is determined by its signless Laplacian permanental polynomial;
• $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$29 is determined by its signless Laplacian permanental polynomial (Wu et al., 2022).

The proofs combine coefficient identities, recurrence formulas, coalescence formulas, transformed explicit expressions, and special evaluations at $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$30. A central structural fact is that both families are nearly regular with degree sequence

$\pi(L(G);x)=\operatorname{per}(xI-L(G)),$31

and the paper uses the lemma that if $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$32 is nearly regular and

$\pi(L(G);x)=\operatorname{per}(xI-L(G)),$33

then $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$34 has the same degree sequence as $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$35 (Wu et al., 2022). This sharply restricts the possible copermanental mates.

The internal uniqueness step compares transformed expressions $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$36 or $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$37. The largest exponents in those expressions depend linearly on $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$38, so equality of polynomials forces equality of the exponent patterns, which in turn determines $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$39, up to the symmetry $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$40 in $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$41 (Wu et al., 2022). Disconnected competitors are then ruled out by the evaluations

$\pi(L(G);x)=\operatorname{per}(xI-L(G)),$42

together with explicit $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$43 values for the target families (Wu et al., 2022).

The same paper also notes a relation of Faria: for a bipartite graph $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$44,

$\pi(L(G);x)=\operatorname{per}(xI-L(G)),$45

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 (Wu et al., 2022).

6. Structured families, later developments, and open problems

The 2025 paper (Mitra et al., 17 Sep 2025) studies several further families: coconut trees $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$46, regular spider trees $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$47, perfect binary trees $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$48, and corona products such as $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$49, $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$50, and $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$51. It defines the same two graph polynomials

$\pi(L(G);x)=\operatorname{per}(xI-L(G)),$52

and frames the central question as whether a graph is determined by one polynomial, by the other, or only by the pair (Mitra et al., 17 Sep 2025).

For bipartite graphs, the paper repeatedly uses

$\pi(L(G);x)=\operatorname{per}(xI-L(G)),$53

so one calculation yields both polynomials (Mitra et al., 17 Sep 2025). This simplification underlies the treatment of coconut trees, spider trees, and perfect binary trees.

Several exact polynomial recurrences are obtained. For the regular spider $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$54,

$\pi(L(G);x)=\operatorname{per}(xI-L(G)),$55

where

$\pi(L(G);x)=\operatorname{per}(xI-L(G)),$56

For the perfect binary tree $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$57,

$\pi(L(G);x)=\operatorname{per}(xI-L(G)),$58

with

$\pi(L(G);x)=\operatorname{per}(xI-L(G)),$59

These formulas provide general computation schemes, but the characterization theorems proved in the paper are more limited (Mitra et al., 17 Sep 2025).

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 $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$60 and $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$61,
• the spider $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$62,
• the perfect binary trees $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$63 and $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$64,
• $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$65, $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$66, and $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$67,
• $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$68,
• $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$69 (Mitra et al., 17 Sep 2025).

At the same time, the scope limitations are explicit. The case $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$70 for $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$71 remains open; the paper gives a recursive formula for all perfect binary trees $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$72, but proves characterization only for $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$73 and $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$74; and it states the conjecture that for $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$75, $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$76 is determined by its (signless) Laplacian permanental polynomials (Mitra et al., 17 Sep 2025). 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 (Mitra et al., 17 Sep 2025).

A further textual issue is documented in the synthesis: some displayed formulas in the manuscript contain typographical inconsistencies, especially in the $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$77, $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$78, and $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$79 sections (Mitra et al., 17 Sep 2025). 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 (Wu et al., 2022, Mitra et al., 17 Sep 2025); it admits recursive and block-decomposition methods through weighted digraphs, cut-vertices, and $\pi(L(G);x)=\operatorname{per}(xI-L(G)),$80-partitions (Singh et al., 2017); and it fits naturally into the generalized permanent framework

$\pi(L(G);x)=\operatorname{per}(xI-L(G)),$81

where Laplacian and signless Laplacian specializations are immediate but technically more resistant to edge-reconstruction arguments than the determinant-based Laplacian characteristic polynomial (Zhang et al., 2023). This suggests that the subject sits at the intersection of permanental combinatorics, graph reconstruction, and graph characterization by polynomial invariants.