Signless Laplacian Permanental Polynomial

• The signless Laplacian permanental polynomial is a graph invariant defined as the permanent of (xI - Q(G)) that encodes key parameters like vertex count, edge count, and triangle frequencies.
• Its coefficients capture structural details such as degrees and cycle counts, enabling the precise characterization of families like bicyclic graphs.
• Researchers apply recursive formulas and deletion techniques to compute these polynomials and investigate graph reconstruction from edge-deleted subgraphs.

Searching arXiv for recent and foundational papers on signless Laplacian permanental polynomials and related reconstruction/characterization results. The signless Laplacian permanental polynomial of a simple undirected graph $G$ with degree matrix $D(G)$, adjacency matrix $A(G)$, and signless Laplacian matrix $Q(G)=D(G)+A(G)$ is the polynomial obtained by taking the permanent of $xI-Q(G)$. In the notation of different papers, it appears as $\pi(Q(G);x)$, $\psi(Q(G);x)$, or $o_4(G;x)$, each defined by $\mathrm{per}(xI-Q(G))$ (Wu et al., 2022). It is studied as a graph invariant with two closely related aims: to determine whether it distinguishes nonisomorphic graphs, and to understand how much structural information can be recovered from the polynomial itself or from collections of polynomials of graph minors such as edge-deleted subgraphs (Zhang et al., 2023).

1. Definition and algebraic setting

Let $G$ be a graph on $D(G)$0 vertices. The permanent of an $D(G)$1 matrix $D(G)$2 is

$D(G)$3

where the sum ranges over all permutations of $D(G)$4 (Wu et al., 2022). With $D(G)$5, the signless Laplacian permanental polynomial is

$D(G)$6

This construction parallels several more familiar graph polynomials. It is formally analogous to the signless Laplacian characteristic polynomial $D(G)$7, but it uses the permanent rather than the determinant. That distinction is substantial: the permanent lacks the alternating-sign cancellation of the determinant, and the resulting polynomial behaves differently in both computation and reconstruction theory (Zhang et al., 2023).

The literature represented here studies the signless Laplacian permanental polynomial together with the Laplacian permanental polynomial $D(G)$8, where $D(G)$9. One recurring fact is that for bipartite graphs,

$A(G)$0

which collapses the distinction between the Laplacian and signless Laplacian permanental settings for trees and other bipartite families (Mitra et al., 17 Sep 2025).

2. Encoded graph data and coefficient information

A central reason for studying $A(G)$1 is that its coefficients encode basic graph parameters. If

$A(G)$2

then the cited results give

$A(G)$3

$A(G)$4

and

$A(G)$5

where $A(G)$6, $A(G)$7 are the vertex degrees, and $A(G)$8 is the number of triangles (Mitra et al., 17 Sep 2025).

Consequently, $A(G)$9 determines $Q(G)=D(G)+A(G)$0, $Q(G)=D(G)+A(G)$1, $Q(G)=D(G)+A(G)$2, and $Q(G)=D(G)+A(G)$3 (Mitra et al., 17 Sep 2025). In proofs of graph determination, these coefficient identities are used to constrain possible degree sequences and triangle counts. The 2022 bicyclic-graph paper states that its arguments exploit the way coefficients encode the degree sequence, the numbers of vertices and edges, and certain subgraph counts such as cycles and triangles, and it cites this coefficient-level control as a key ingredient of the characterization theorems (Wu et al., 2022).

This coefficient information does not imply that $Q(G)=D(G)+A(G)$4 is a complete invariant for all graphs. Rather, it provides enough structural rigidity to settle specific families. A plausible implication is that the polynomial is particularly effective on classes where degree distributions and small-cycle structure already impose strong global constraints.

3. Determination of bicyclic graphs

A major characterization result concerns two standard classes of bicyclic graphs. The paper (Wu et al., 2022) studies

• $Q(G)=D(G)+A(G)$5: the graph formed by joining cycles $Q(G)=D(G)+A(G)$6 and $Q(G)=D(G)+A(G)$7 at the ends of a path $Q(G)=D(G)+A(G)$8;
• $Q(G)=D(G)+A(G)$9: the graph consisting of two given vertices joined by three disjoint paths of orders $xI-Q(G)$0, with at most one of $xI-Q(G)$1 equal to zero.

Its main theorems establish that both classes are determined by their Laplacian permanental polynomial and also by their signless Laplacian permanental polynomial (Wu et al., 2022). In the signless Laplacian formulation, if

$xI-Q(G)$2

then $xI-Q(G)$3, and similarly for $xI-Q(G)$4 (Wu et al., 2022). Equivalently, within these classes there are no nonisomorphic cospectral graphs with respect to the signless Laplacian permanental polynomial.

The proofs combine explicit polynomial calculations with structural uniqueness arguments. The paper states that it uses leading-term analysis, evaluation at special points such as $xI-Q(G)$5, coefficient inspection, recursive and combinatorial deletion formulas, and a uniqueness lemma for nearly regular graphs (Wu et al., 2022). It also gives explicit recursive and closed-form expressions for the relevant permanental polynomials and special-value formulas for $xI-Q(G)$6 and $xI-Q(G)$7 at $xI-Q(G)$8 (Wu et al., 2022).

These results are notable because they answer, for the two families considered, the characterization problem in its strongest form: equality of the signless Laplacian permanental polynomial forces isomorphism. The same paper states that this closes an open problem posed in earlier work by Wu et al. for these bicyclic classes (Wu et al., 2022).

4. Computational recurrences and explicit families

The computation of $xI-Q(G)$9 is developed through recursive expansions on vertices, edges, and cycle structure. One formula states that if $\pi(Q(G);x)$0 is a vertex, $\pi(Q(G);x)$1 its neighbors, and $\pi(Q(G);x)$2 the set of cycles through $\pi(Q(G);x)$3, then

$\pi(Q(G);x)$4

with analogous notation for principal submatrices obtained by deletions (Mitra et al., 17 Sep 2025). An edge-based recursion is also given, incorporating edge deletion, endpoint deletion, and contributions from cycles containing the edge (Mitra et al., 17 Sep 2025).

These recurrences support explicit formulas for several structured graph families. The same source emphasizes recursive expansion, exploitation of symmetry, block and tridiagonal recurrences, lattice decompositions for corona products, and coefficient analysis as the main computational techniques (Mitra et al., 17 Sep 2025).

Graph family	Result for $\pi(Q(G);x)$5	Status of determination
Coconut tree $\pi(Q(G);x)$6, $\pi(Q(G);x)$7	Explicit formulas given	Determined
Regular spider $\pi(Q(G);x)$8	$\pi(Q(G);x)$9	Determined for $\psi(Q(G);x)$0
Perfect binary tree $\psi(Q(G);x)$1	$\psi(Q(G);x)$2	Determined for $\psi(Q(G);x)$3
$\psi(Q(G);x)$4	Explicit formula via lattice expansion	Determined for $\psi(Q(G);x)$5
$\psi(Q(G);x)$6, $\psi(Q(G);x)$7	Explicit formulas given	Determined for all $\psi(Q(G);x)$8

For regular spiders, the formula

$\psi(Q(G);x)$9

is given with $o_4(G;x)$0, $o_4(G;x)$1, and $o_4(G;x)$2 (Mitra et al., 17 Sep 2025). For perfect binary trees,

$o_4(G;x)$3

where $o_4(G;x)$4, $o_4(G;x)$5, and $o_4(G;x)$6 for $o_4(G;x)$7 (Mitra et al., 17 Sep 2025).

The paper further reports that $o_4(G;x)$8 and $o_4(G;x)$9, $\mathrm{per}(xI-Q(G))$0, $\mathrm{per}(xI-Q(G))$1, $\mathrm{per}(xI-Q(G))$2, $\mathrm{per}(xI-Q(G))$3 for $\mathrm{per}(xI-Q(G))$4, and $\mathrm{per}(xI-Q(G))$5, $\mathrm{per}(xI-Q(G))$6 are determined by their signless Laplacian permanental polynomial (Mitra et al., 17 Sep 2025). For the corona family $\mathrm{per}(xI-Q(G))$7, characterization for larger $\mathrm{per}(xI-Q(G))$8 is stated as conjectural rather than settled (Mitra et al., 17 Sep 2025).

5. Edge reconstruction and deleted-subgraph data

The reconstruction problem asks whether $\mathrm{per}(xI-Q(G))$9 can be recovered from data attached to graph deletions. For the notation $G$0, the reconstruction formula

$G$1

is established for a simple graph with $G$2 vertices and $G$3 edges (Zhang et al., 2023). Here $G$4 denotes deletion of the edge $G$5, and $G$6 deletion of both endpoints.

From this, the paper proves that if $G$7, then the permanental polynomial $G$8, and in particular the signless Laplacian permanental polynomial when defined in this way, can be reconstructed from the multiset

$G$9

(Zhang et al., 2023). It also states that standard recurrence methods together with an initial condition determined from the same data yield uniqueness when $D(G)$00 (Zhang et al., 2023).

A key negative result is that reconstruction from edge-deleted subgraphs alone is not available in general for the signless Laplacian permanental polynomial (Zhang et al., 2023). This sharply contrasts with the signless Laplacian characteristic polynomial $D(G)$01, for which

$D(G)$02

so the edge-deleted deck alone suffices when $D(G)$03 (Zhang et al., 2023). The distinction is often overlooked: permanental and characteristic reconstruction theories are not parallel here.

When $D(G)$04, the cited paper states that the corresponding reconstruction problem for $D(G)$05 remains open (Zhang et al., 2023). This unresolved case is structurally significant because many bicyclic graphs satisfy $D(G)$06, whereas unicyclic graphs satisfy $D(G)$07; the existing theorem therefore separates these regimes cleanly.

6. Scope, limitations, and research directions

The current theory supports two broad conclusions. First, the signless Laplacian permanental polynomial has substantial discriminative power on carefully structured graph families. This is explicit in the bicyclic classes $D(G)$08 and $D(G)$09 (Wu et al., 2022) and in several tree-like and corona families studied later (Mitra et al., 17 Sep 2025). Second, this power is still family-dependent rather than universal: the papers repeatedly formulate the central task as determining which graphs are characterized by their Laplacian or signless Laplacian permanental polynomials (Mitra et al., 17 Sep 2025).

Several limitations are stated directly in the cited works. The 2022 bicyclic-graph paper notes that for general graphs it remains open to determine for which families the Laplacian and signless Laplacian permanental polynomials are complete invariants, including arbitrary bicyclic graphs and larger classes (Wu et al., 2022). The 2023 reconstruction paper leaves open the case $D(G)$10 for permanental reconstruction from edge-deleted and two-vertex-deleted data (Zhang et al., 2023). The 2025 paper provides explicit formulas for larger corona families and conjectures determination beyond the cases it proves, particularly for $D(G)$11 with $D(G)$12 (Mitra et al., 17 Sep 2025).

A further point of interpretation concerns bipartite graphs. Because $D(G)$13 for bipartite $D(G)$14, results proved in the signless Laplacian permanental setting for trees are simultaneously Laplacian results (Mitra et al., 17 Sep 2025). Outside the bipartite case, however, the two invariants diverge. This suggests that signless Laplacian permanental methods are especially informative for families where odd-cycle structure plays a role, while still retaining exact coincidence with the Laplacian theory on bipartite subclasses.

In this state of the literature, the signless Laplacian permanental polynomial occupies a position between a computable symbolic invariant and a family-sensitive characterization tool. The available results show that it can recover concrete structural data, support explicit recurrences, determine important graph families up to isomorphism, and participate in nontrivial reconstruction theory, while leaving open the general classification problem and several boundary cases (Wu et al., 2022).