Line Graph Laplacian Correlation

• Line Graph Laplacian Correlation is a method defining Laplacian coefficients of forests through closed-walk counts in their line graphs.
• The approach uses Newton identities to derive explicit formulas for coefficients cₙ₋ₖ (1 ≤ k ≤ 6) based on combinatorial walk data.
• It provides a unified framework linking spectral invariants with adjacency-based measures, with practical illustrations using examples like P₆.

Line graph Laplacian correlation refers to the explicit relationship between the Laplacian coefficients of a forest and the closed-walk combinatorics of its line graph. This concept centers around expressing the so-called "small" Laplacian coefficients $c_{n-k}$, for $1 \leq k \leq 6$, of a forest $F$ in terms of closed walks in $F$ and its line graph $L(F)$. The main result is a concrete set of formulas deriving Laplacian spectral invariants of a forest directly from elementary adjacency walk counts, particularly in the line graph, illuminating a deep combinatorial bridge between these objects (Ghalavand et al., 2021).

1. The Laplacian Polynomial and Its Combinatorial Interpretation

Given an $n$-vertex simple graph $G$ with Laplacian matrix $L(G)$ and eigenvalues $0=\lambda_1 \leq \cdots \leq \lambda_n$, the Laplacian polynomial is

$$\psi(G,\lambda) = \det(\lambda I_n - L(G)) = \sum_{k=0}^n (-1)^{n-k} c_k \lambda^k.$$

The coefficients $1 \leq k \leq 6$0 are the elementary symmetric functions of the Laplacian eigenvalues. For a forest $1 \leq k \leq 6$1, it is established that $1 \leq k \leq 6$2, where $1 \leq k \leq 6$3 denotes the number of $1 \leq k \leq 6$4-matchings in the subdivision graph $1 \leq k \leq 6$5. Alternatively, $1 \leq k \leq 6$6 can be regarded as a weighted count of spanning subforests of size $1 \leq k \leq 6$7, but the $1 \leq k \leq 6$8-matching interpretation is the most direct in this setting (Ghalavand et al., 2021).

2. Newton Identities and Laplacian Coefficients

For any graph $1 \leq k \leq 6$9, the Newton identities connect the coefficients $F$0 to traces of powers of the Laplacian,

$F$1

Specifically for $F$2, and $F$3: \begin{align*} c_{n-4} &= \frac{1}{24}\left(\mu_1^4 - 6\mu_1^2\mu_2 + 3\mu_2^2 + 8\mu_1\mu_3 - 6\mu_4\right), \ c_{n-5} &= \frac{1}{120}\left( \mu_1^5 - 10\mu_1^3\mu_2 + 5\mu_1\mu_2^2 + 15\mu_1^2\mu_3 - 20\mu_2\mu_3 -20\mu_1\mu_4 + 24\mu_5 \right), \ c_{n-6} &= \frac{1}{720}\left( \mu_1^6 - \cdots -120\mu_6 \right), \end{align*} where $F$4 with $F$5 the number of edges and higher traces $F$6 are to be computed in terms of closed walks (Ghalavand et al., 2021).

3. Closed Walk Representations in Forests and Line Graphs

Powers of $F$7 can be expanded in the degree matrix $F$8 and adjacency matrix $F$9, with monomials associated to closed walks. In forests, all mixed terms admit a combinatorial reinterpretation as closed walks in either $F$0 or its line graph $F$1. The number of closed walks of length $F$2 in a graph $F$3 is $F$4. Key results for a forest $F$5 (edges $F$6, line graph $F$7) are:

• $F$8, with $F$9
• $L(F)$0
• $L(F)$1

Analogous explicit formulas exist for $L(F)$2 and $L(F)$3 as linear combinations of walk counts $L(F)$4 and various structural quantities of $L(F)$5 and its iterates. Notably, in forests, odd-length closed walks beyond $L(F)$6 vanish due to the acyclicity of $L(F)$7 (Ghalavand et al., 2021).

4. Explicit Closed-Walk Formulas for $L(F)$8, $L(F)$9, $n$0

By substituting the closed-walk expansions of $n$1 into the Newton identities, the following explicit forms are obtained: $n$2 with $n$3 and $n$4 given by similar explicit (but lengthier) combinations of walk counts $n$5, numbers of edges in iterated line graphs $n$6, $n$7, triangle counts $n$8, etc. Each term $n$9 is $G$0, and the only appearance of closed walks in the original forest $G$1 is through $G$2. The full expansions are detailed in Theorem 3.5 of (Ghalavand et al., 2021).

5. Representative Computation: The Path $G$3

For the path $G$4 ($G$5, $G$6), the line graph $G$7 is $G$8. Direct calculation yields:

• $G$9
• $L(G)$0
• $L(G)$1
• $L(G)$2
• $L(G)$3

Substituting these values gives $L(G)$4, coinciding with the Wiener index. Similarly, $L(G)$5 and $L(G)$6, matching known structural results for trees (Ghalavand et al., 2021).

6. Scope, Extensions, and Structural Significance

These closed-walk formulas confirm that the Laplacian spectral information (elementary symmetric functions of eigenvalues) for forests is determined by combinatorial data captured in the adjacency spectra of the forest's line graph. Only the trivial back-and-forth walk counts $L(G)$7 in $L(G)$8 enter, with all higher closed-walk terms arising from $L(G)$9 and its iterates. This result bridges the Laplacian coefficients of $0=\lambda_1 \leq \cdots \leq \lambda_n$0 and adjacency-based closed-walk data in $0=\lambda_1 \leq \cdots \leq \lambda_n$1. An explicit bijective counting argument justifies each term's appearance in the expansions of $0=\lambda_1 \leq \cdots \leq \lambda_n$2.

Extensions include attempts to generalize these formulas to $0=\lambda_1 \leq \cdots \leq \lambda_n$3 and beyond, acknowledging rapid combinatorial growth, and investigations regarding their validity or necessary modifications in graphs with a few short cycles. For such graphs, walk counts $0=\lambda_1 \leq \cdots \leq \lambda_n$4 for odd $0=\lambda_1 \leq \cdots \leq \lambda_n$5 may become nonzero, affecting the resulting expressions (Ghalavand et al., 2021).

7. Summary Table: Main Quantities and Their Origins

Quantity	Graph	Description
$0=\lambda_1 \leq \cdots \leq \lambda_n$6	$0=\lambda_1 \leq \cdots \leq \lambda_n$7	$0=\lambda_1 \leq \cdots \leq \lambda_n$8-th Laplacian coefficient
$0=\lambda_1 \leq \cdots \leq \lambda_n$9	$$\psi(G,\lambda) = \det(\lambda I_n - L(G)) = \sum_{k=0}^n (-1)^{n-k} c_k \lambda^k.$$0	Closed walks of length $$\psi(G,\lambda) = \det(\lambda I_n - L(G)) = \sum_{k=0}^n (-1)^{n-k} c_k \lambda^k.$$1 in $$\psi(G,\lambda) = \det(\lambda I_n - L(G)) = \sum_{k=0}^n (-1)^{n-k} c_k \lambda^k.$$2
$$\psi(G,\lambda) = \det(\lambda I_n - L(G)) = \sum_{k=0}^n (-1)^{n-k} c_k \lambda^k.$$3	$$\psi(G,\lambda) = \det(\lambda I_n - L(G)) = \sum_{k=0}^n (-1)^{n-k} c_k \lambda^k.$$4	Closed walks of length $$\psi(G,\lambda) = \det(\lambda I_n - L(G)) = \sum_{k=0}^n (-1)^{n-k} c_k \lambda^k.$$5 in line graph
$$\psi(G,\lambda) = \det(\lambda I_n - L(G)) = \sum_{k=0}^n (-1)^{n-k} c_k \lambda^k.$$6	$$\psi(G,\lambda) = \det(\lambda I_n - L(G)) = \sum_{k=0}^n (-1)^{n-k} c_k \lambda^k.$$7	Edges in $$\psi(G,\lambda) = \det(\lambda I_n - L(G)) = \sum_{k=0}^n (-1)^{n-k} c_k \lambda^k.$$8
$$\psi(G,\lambda) = \det(\lambda I_n - L(G)) = \sum_{k=0}^n (-1)^{n-k} c_k \lambda^k.$$9	$1 \leq k \leq 6$00	Triangles in $1 \leq k \leq 6$01
$1 \leq k \leq 6$02	$1 \leq k \leq 6$03	Edges in $1 \leq k \leq 6$04-th iterated line graph

All main Laplacian coefficient formulas for forests in this framework are ultimately algorithmic combinations of these quantities, providing an intrinsic connection between their Laplacian and adjacency walk structures (Ghalavand et al., 2021).