[Paper Review] Laplacian Estrada index of trees

Aleksandar Ilić, Bo Zhou|arXiv (Cornell University)|Jun 15, 2011

Graph theory and applicationsMathematics23 references22 citations

TL;DR

This paper establishes that among all trees with $ n $ vertices, the path $ P_n $ minimizes and the star $ S_n $ maximizes the Laplacian Estrada index (LEE), with the double star $ S_n(2,n-2) $ being the unique tree with the second-highest LEE. Using a spectral transformation ($ \sigma $-transform) and a known relation between the Laplacian Estrada index and the Estrada index of a line graph, the authors prove the ordering via spectral moment comparisons and root analysis of characteristic polynomials.

ABSTRACT

Let $G$ be a simple graph with $n$ vertices and let $μ_1 \geqslant μ_2 \geqslant...\geqslant μ_{n - 1} \geqslant μ_n = 0$ be the eigenvalues of its Laplacian matrix. The Laplacian Estrada index of a graph $G$ is defined as $LEE (G) = \sum\limits_{i = 1}^n e^{μ_i}$. Using the recent connection between Estrada index of a line graph and Laplacian Estrada index, we prove that the path $P_n$ has minimal, while the star $S_n$ has maximal $LEE$ among trees on $n$ vertices. In addition, we find the unique tree with the second maximal Laplacian Estrada index.

Motivation & Objective

• To determine the extremal trees (minimal and maximal) for the Laplacian Estrada index (LEE) among all $ n $-vertex trees.
• To validate LEE as a topological index for measuring branching in alkanes by verifying it satisfies the expected ordering: $ LEE(P_n) < LEE(T) < LEE(S_n) $ for all non-path, non-star trees $ T $.
• To identify the unique tree with the second-highest LEE among $ n $-vertex trees.
• To establish a spectral connection between the Laplacian Estrada index of a tree and the Estrada index of its line graph, enabling the use of known results on Estrada indices.

Proposed method

• Utilizes the known identity $ LEE(G) = n - m + e^2 \cdot EE(\mathcal{L}(G)) $ for bipartite graphs, where $ \mathcal{L}(G) $ is the line graph of $ G $.
• Applies the spectral moment method: $ EE(G) = \sum_{k=0}^\infty \frac{M_k(G)}{k!} $, where $ M_k(G) $ counts closed walks of length $ k $.
• Employs the $ \sigma $-transform: a graph operation that relocates pendent edges from a high-degree vertex to its neighbor, increasing spectral moments.
• Uses the fact that $ EE(H) > EE(G) $ if $ M_k(H) \geq M_k(G) $ for all $ k $, with strict inequality for some $ k $, to compare line graphs.
• Analyzes the characteristic polynomial of the Laplacian matrix of double stars $ S_n(a,b) $, focusing on the three nontrivial roots to compare $ LEE $ values.
• Applies root localization techniques to bound the largest, middle, and smallest positive roots of the cubic factor of the Laplacian characteristic polynomial.

Experimental results

Research questions

• RQ1 Which tree on $ n $ vertices has the minimal Laplacian Estrada index?
• RQ2 Which tree on $ n $ vertices has the maximal Laplacian Estrada index?
• RQ3 What is the unique tree with the second-highest Laplacian Estrada index among $ n $-vertex trees?
• RQ4 Does the Laplacian Estrada index satisfy the expected ordering $ LEE(P_n) < LEE(T) < LEE(S_n) $ for all $ n $-vertex trees $ T \neq P_n, S_n $?
• RQ5 Can the $ \sigma $-transform be used to prove monotonicity of LEE under structural changes in trees?

Key findings

• The path $ P_n $ has the minimal Laplacian Estrada index among all $ n $-vertex trees.
• The star $ S_n $ has the maximal Laplacian Estrada index among all $ n $-vertex trees.
• The double star $ S_n(2,n-2) $ is the unique tree with the second-highest Laplacian Estrada index for $ n \geq 5 $.
• For $ n \geq 8 $, $ LEE(S_n(3,n-3)) > LEE(S_n(\lfloor n/2\rfloor, \lceil n/2\rfloor)) $, confirming $ S_n(2,n-2) $ is second-largest.
• The $ \sigma $-transform strictly increases the Laplacian Estrada index in bipartite graphs, including trees, due to increased spectral moments in their line graphs.
• For $ n \geq 6 $, the tree $ C_n(n-5) $, formed by attaching $ n-5 $ vertices to the center of a $ P_5 $, is the unique tree with the fourth-highest LEE, based on computational verification up to $ n = 22 $.