SAMS Subject Classification Number: 6

Let \(G\) be a graph with \(n\) vertices. The energy \(E_n(G)\) of \(G\) is defined as the sum of the absolute values of its eigenvalues. The Hosoya index \(Z(G)\) of \(G\) is the number of independent edge subsets of \(G\). And the Merrifield-Simmons index \(\sigma (G)\) is the number of independent vertex subsets of \(G\), including the empty set. The three above-mentioned closely related graph invariants are studied in mathematical chemistry.

The talk will be discussing extremal caterpillars, relative to the Energy, Hosoya index and Merrifield-Simmons index. Since the Energy and Hosoya index can be redefined in terms of an auxiliary invariant, \(\displaystyle{M(G,x)=\sum_{k\geq0}m(G,k)x^{k}}\), where \(m(G,k)\) is the number of independent edge subsets of order \(k\) in \(G\), we focus on finding extremal caterpillars, relative to \(M(.,x)\). We use similar methods as with \(M(.,x)\) to conclude on extremal caterpillars, relative to the Merrifield-Simmons index.