Complete solution to a conjecture on the maximal energy of unicyclic graphs

TL;DR

This paper proves a long-standing conjecture about which unicyclic graphs have maximal energy, using advanced mathematical techniques, and corrects the conjecture for small graph sizes.

Contribution

It provides a complete proof of the conjecture on maximal energy unicyclic graphs and corrects it for small cases using Coulson integral formula and combinatorial methods.

Findings

Confirmed the conjecture for most n, except n=4

Identified P_4^3 as the maximal energy unicyclic graph for n=4

Established a complete characterization of extremal unicyclic graphs

Abstract

For a given simple graph $G$, the energy of $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let $P_n^{\ell}$ be the unicyclic graph obtained by connecting a vertex of $C_\ell$ with a leaf of $P_{n-\ell}$\,. In [G. Caporossi, D. Cvetkovi\'c, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with extremal energy, {\it J. Chem. Inf. Comput. Sci.} {\bf 39}(1999) 984--996], Caporossi et al. conjectured that the unicyclic graph with maximal energy is $C_n$ if $n\leq 7$ and $n=9,10,11,13,15$\,, and $P_n^6$ for all other values of $n$. In this paper, by employing the Coulson integral formula and some knowledge of real analysis, especially by using certain combinatorial technique, we completely solve this conjecture. However, it turns out that for $n=4$ the conjecture is not true, and $P_4^3$ should be the unicyclic graph with maximal energy.