[Paper Review] On the Wiener index and Laplacian coefficients of graphs with given diameter or radius

Aleksandar Ilić, Andreja Ilić+1 more|arXiv (Cornell University)|May 29, 2011

Graph theory and applicationsMathematics17 references20 citations

TL;DR

This paper characterizes trees and connected graphs with given diameter or radius that minimize all Laplacian coefficients simultaneously, using a connection between Laplacian coefficients and matching numbers in subdivided graphs. The key result is that the caterpillar $ C_{n,d} $ minimizes all coefficients among $ n $-vertex trees of diameter $ d $, and $ C_{n,2r-1} $ does so among graphs of radius $ r $, generalizing prior results on Wiener index minimization.

ABSTRACT

Let $G$ be a simple undirected $n$-vertex graph with the characteristic polynomial of its Laplacian matrix $L(G)$, $\det (λI - L (G))=\sum_{k = 0}^n (-1)^k c_k λ^{n - k}$. It is well known that for trees the Laplacian coefficient $c_{n-2}$ is equal to the Wiener index of $G$. Using a result of Zhou and Gutman on the relation between the Laplacian coefficients and the matching numbers in subdivided bipartite graphs, we characterize first the trees with given diameter and then the connected graphs with given radius which simultaneously minimize all Laplacian coefficients. This approach generalizes recent results of Liu and Pan [MATCH Commun. Math. Comput. Chem. 60 (2008), 85--94] and Wang and Guo [MATCH Commun. Math. Comput. Chem. 60 (2008), 609--622] who characterized $n$-vertex trees with fixed diameter $d$ which minimize the Wiener index. In conclusion, we illustrate on examples with Wiener and modified hyper-Wiener index that the opposite problem of simultaneously maximizing all Laplacian coefficients has no solution.

Motivation & Objective

• To extend prior results on Wiener index minimization in trees with fixed diameter to the simultaneous minimization of all Laplacian coefficients.
• To characterize the structure of connected $ n $-vertex graphs with fixed radius $ r $ that minimize all Laplacian coefficients.
• To generalize results of Liu and Pan and Wang and Guo by proving that $ C_{n,d} $ minimizes all $ c_k $ coefficients among trees of diameter $ d $.
• To demonstrate that the problem of maximizing all Laplacian coefficients simultaneously has no solution, using counterexamples with Wiener and modified hyper-Wiener indices.

Proposed method

• Utilizes the identity $ c_k(T) = m_k(S(T)) $, where $ c_k $ is the $ k $-th Laplacian coefficient of a tree $ T $, and $ m_k(S(T)) $ is the number of $ k $-matchings in the subdivision graph $ S(T) $.
• Applies Lemma 2.2 to show that matching numbers in path unions are minimized when path lengths are as equal as possible, used to compare matching counts in caterpillar subdivision graphs.
• Constructs a correspondence between matchings in the subdivision of a general caterpillar $ C(a_1,\dots,a_{d-1}) $ and those in $ C_{n,d} $, showing $ m_k(S(C_{n,d})) \leq m_k(S(C(a_1,\dots,a_{d-1}))) $.
• Uses interlacing of Laplacian eigenvalues to show that edge deletion does not increase Laplacian coefficients, enabling reduction from general graphs to spanning trees.
• Applies the result that $ c_k(C_{n,2r-1}) \leq c_k(C_{n,2r}) $ to show $ C_{n,2r-1} $ minimizes coefficients among graphs of radius $ r $.
• Employs computational verification on trees up to 20 vertices to analyze extremal graphs for maximizing $ c_k $, particularly for $ c_{n-2} $ (Wiener index) and $ c_{n-3} $ (modified hyper-Wiener index).

Experimental results

Research questions

• RQ1 Which $ n $-vertex trees of fixed diameter $ d $ minimize all Laplacian coefficients $ c_k $ simultaneously?
• RQ2 Which connected $ n $-vertex graphs of fixed radius $ r $ minimize all Laplacian coefficients $ c_k $ simultaneously?
• RQ3 Can the problem of maximizing all Laplacian coefficients simultaneously be solved, and if not, why?
• RQ4 How do the extremal graphs for Wiener index and modified hyper-Wiener index compare under fixed diameter or radius?
• RQ5 What structural properties of caterpillars ensure minimal Laplacian coefficients across all $ k $?

Key findings

• The caterpillar $ C_{n,d} $ is the unique $ n $-vertex tree of diameter $ d $ that minimizes all Laplacian coefficients $ c_k $ for $ k = 0,1,\dots,n $.
• For connected $ n $-vertex graphs of radius $ r $, the caterpillar $ C_{n,2r-1} $ minimizes all Laplacian coefficients $ c_k $.
• The Wiener index $ c_{n-2} $ and modified hyper-Wiener index $ c_{n-3} $ are minimized by $ C_{n,d} $ and $ C_{n,2r-1} $, respectively, among their respective graph classes.
• There is no single graph that maximizes all Laplacian coefficients simultaneously, as extremal graphs for $ c_{n-2} $ and $ c_{n-3} $ differ even for small $ n $.
• For $ n=18 $, diameter $ d=4 $, the tree maximizing $ c_{16} $ (Wiener index) is distinct from the one maximizing $ c_{15} $ (modified hyper-Wiener index).
• For $ n=17 $, radius $ r=5 $, the tree maximizing $ c_{15} $ (Wiener index) is different from the one maximizing $ c_{14} $ (modified hyper-Wiener index).