On the minimal energy of conjugated unicyclic graphs with maximum degree at most 3

The energy of a graph GG, denoted by E(G)E(G), is defined as the sum of the absolute values of all eigenvalues of GG. Let nn be an even number and UnUn be the set of all conjugated unicyclic graphs of order nn with maximum degree at most 3. Let Snn2 be the radialene graph obtained by attaching a pendant edge to each vertex of the cycle Cn2. Cao et al. (2009) showed that if n≥8n≥8, Snn2≇G∈Un and the girth of GG is not divisible by 4, then E(G)>E(Snn2). Let AnAn be the unicyclic graph obtained by attaching a 4-cycle to one of the two leaf vertices of the path Pn2−1 and a pendent edge to each other vertices of Pn2−1. In this paper, we prove that AnAn is the unique unicyclic graph in UnUn with minimal energy for n≥8n≥8.