[Paper Review] Resolvent Energy of Unicyclic, Bicyclic and Tricyclic Graphs

Luiz Emílio Allem, Juliane Capaverde+4 more|arXiv (Cornell University)|Dec 30, 2015

Ferrocene Chemistry and Applications2 references8 citations

TL;DR

This paper confirms conjectures on the resolvent energy of unicyclic, bicyclic, and tricyclic graphs by analyzing spectral moments and characteristic polynomials. It proves that the unicyclic graph $X_n$ (a triangle with $n-3$ pendent vertices) maximizes resolvent energy, while $\tilde{X}_n$ (a 4-cycle with $n-4$ pendants) does so among bipartite unicyclic graphs; similarly, $Z_n^1$ is shown to maximize resolvent energy among tricyclic graphs of order $n$. The results are derived using eigenvalue-based energy formulas and characteristic polynomial differentiation.

ABSTRACT

The resolvent energy of a graph $G$ of order $n$ is defined as $ER=\sum_{i=1}^n (n-λ_i)^{-1}$, where $λ_1,λ_2,\ldots,λ_n$ are the eigenvalues of $G$. In a recent work [Gutman et al., {\it MATCH Commun. Math. Comput. Chem.\/} {\bf 75} (2016) 279--290] the structure of the graphs extremal w.r.t. $ER$ were conjectured, based on an extensive computer--aided search. We now confirm the validity of some of these conjectures.

Motivation & Objective

• Identify the unicyclic, bicyclic, and tricyclic graphs with maximum and minimum resolvent energy based on prior conjectures.
• Confirm that the resolvent energy of a graph is maximized by specific extremal structures through spectral analysis.
• Investigate the role of odd and even cycles in determining resolvent energy extremality in unicyclic graphs.
• Establish a rigorous mathematical foundation for resolvent energy extremality using characteristic polynomials and their derivatives.
• Extend results from the Estrada index to the resolvent energy by adapting proofs based on spectral moments.

Proposed method

• The resolvent energy is defined as $ER(G) = \sum_{i=1}^n \frac{1}{n - \lambda_i}$, where $\lambda_i$ are the eigenvalues of the adjacency matrix.
• Equation (2) expresses resolvent energy as a series involving spectral moments $M_k(G) = \sum_{i=1}^n \lambda_i^k$, enabling comparison via moment dominance.
• Characteristic polynomials $\phi(G, \lambda)$ are derived using Lemma 2, which relates $\phi(G, \lambda)$ to subgraphs via vertex and edge deletions.
• Resolvent energy is computed via $ER(G) = \frac{\phi'(G, n)}{\phi(G, n)}$, allowing direct comparison between graphs using polynomial derivatives.
• Polynomial inequalities are used to prove $ER(G) < ER(H)$ by showing the difference $ER(H) - ER(G)$ has positive numerator and denominator for $n \geq 4$.
• Results from prior works on the Estrada index are adapted to the resolvent energy by replacing $\frac{M_k}{k!}$ with $\frac{M_k}{n^k}$ in series expansions.

Experimental results

Research questions

• RQ1 What unicyclic graph of order $n$ maximizes resolvent energy, and what is its structural form?
• RQ2 How does the presence of odd versus even cycles affect the resolvent energy of unicyclic graphs?
• RQ3 What is the extremal tricyclic graph with maximum resolvent energy, and how is it characterized?
• RQ4 Can spectral moment dominance and characteristic polynomial analysis confirm prior conjectures on resolvent energy extremality?
• RQ5 Is the resolvent energy of $X_n$ strictly greater than that of $\tilde{X}_n$ for all $n \geq 4$?

Key findings

• The unicyclic graph $X_n$, formed by attaching $n-3$ pendent vertices to one vertex of a triangle, maximizes resolvent energy among all unicyclic graphs of order $n$.
• The unicyclic graph $\tilde{X}_n$, formed by attaching $n-4$ pendent vertices to a 4-cycle, maximizes resolvent energy among bipartite unicyclic graphs.
• Among all $n$-vertex tricyclic graphs, $Z_n^1$ has the maximum resolvent energy, with $ER(Z_n^1) > ER(Z_n^i)$ for all $2 \leq i \leq 6$.
• Resolvent energy differences are proven positive via rational functions with positive numerators and denominators for $n \geq 4$, confirming strict inequality.
• The resolvent energy of $X_n$ exceeds that of $\tilde{X}_n$ for all $n \geq 4$, as shown by the positivity of the difference $ER(X_n) - ER(\tilde{X}_n)$.
• Characteristic polynomials of extremal graphs are explicitly computed, enabling exact evaluation of resolvent energy via $ER(G) = \frac{\phi'(G, n)}{\phi(G, n)}$.