The first degcity index \(\operatorname{DC}_{1}(G)\) of a connected graph \(G\) is the edge sum \[\operatorname{DC}_{1}(G)=\sum\limits_{uv\in E(G)}\bigl[e_G(u)+e_G(v)\bigr]\bigl[d_G(u)+d_G(v)\bigr],\] where \(d_G(u)\) and \(e_G(u)\) denote the degree and eccentricity of a vertex \(u\), respectively. The index combines local valency and global distance information in a single degree–eccentricity descriptor. This paper determines closed expressions for the first degcity index under six standard graph operations: disjoint union, join, Cartesian product, composition, symmetric difference and disjunction. The formulas separate the contributions of edges inherited from the factor graphs from the contributions created by the operation. The statements use the eccentricity behaviour in joins and the edge and degree relations in product-type operations, giving formulas that are consistent with the usual definitions of these graph operations.
