Complete solution to a problem on the maximal energy of unicyclic bipartite graphs

TL;DR

This paper proves that among unicyclic bipartite graphs, the graph $P_n^6$ has maximal energy for certain n, resolving an open problem and partially confirming a conjecture about energy maximization.

Contribution

The authors demonstrate that $P_n^6$ has greater energy than $C_n$ for specific n, solving an open problem and advancing understanding of energy in unicyclic bipartite graphs.

Findings

$E(P_n^6)$ exceeds $E(C_n)$ for n=8,12,14 and n≥16

Complete resolution of the open problem regarding energy comparison

Partial confirmation of the conjecture on maximal energy unicyclic graphs

Abstract

The energy of a simple graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Denote by $C_n$ the cycle, and $P_n^{6}$ the unicyclic graph obtained by connecting a vertex of $C_6$ with a leaf of $P_{n-6}$\,. Caporossi et al. conjecture that the unicyclic graph with maximal energy is $P_n^6$ for $n=8,12,14$ and $n\geq 16$. In``Y. Hou, I. Gutman and C. Woo, Unicyclic graphs with maximal energy, {\it Linear Algebra Appl.} {\bf 356}(2002), 27--36", the authors proved that $E(P_n^6)$ is maximal within the class of the unicyclic bipartite $n$-vertex graphs differing from $C_n$\,. And they also claimed that the energy of $C_n$ and $P_n^6$ is quasi-order incomparable and left this as an open problem. In this paper, by utilizing the Coulson integral formula and some knowledge of real analysis, especially by employing certain combinatorial techniques, we show that the energy of $P_n^6$ is greater than that of $C_n$ for $n=8,12,14$ and $n\geq 16$, which completely solves this open problem and partially solves the above conjecture.