On the Sum of the Second Zagreb Index and Its Reciprocal Index for Tricyclic Graphs

For a simple connected graph G, let dG(u) denote the degree of a vertex u. The second Zagreb index Z2(G)=∑uv∈E(G)dG(u)dG(v) and its reciprocal index mZ2(G)=∑uv∈E(G)1/(dG(u)dG(v)) are widely studied topological indices in graph theory. Their sum Z2(G)+Z2m(G) has been studied for trees, unicyclic graphs, and bicyclic graphs, but relevant results for tricyclic graphs remain unexplored. Using the path-lifting transformation, we prove that the maximum value of Z2(G)+Z2m(G) among all tricyclic graphs is Y4=3n3+9n2+58n−763(n−1), which is attained by the extremal graph G4≅S5(n−4,0,0,0). Our work completes the research framework of this index sum from acyclic to tricyclic graphs and provides theoretical support for chemical topological modeling.