2025/12/20 by Julian Allagan, Shawn Langley, Allagan, Julian +5
Engineering · Computer Science · Mathematics · #Structural Analysis and Optimization #Advanced Graph Theory Research #Limits and Structures in Graph Theory
paper · doi:10.48550/arxiv.2512.18378
We study strong r-central 2-trees whose non-central vertices have degrees in \2,3\, focusing on the cases r=1,2,3. For each r, we derive exact degree constraints relating the maximum degree Δ to the numbers of degree-3 and degree-2 tail vertices. In the unicentral case (r=1), we prove that the fan graph is the unique realization for all n≥ 3. For bicentral 2-trees (r=2), we show that the number of degree-3 vertices is always even, establish sharp uniqueness results for x∈\0,2\, prove existence for all feasible values of Δ, and obtain linear lower bounds on the number of non-isomorphic realizations. For tricentral 2-trees (r=3), we characterize extremal configurations, establish a divisibility constraint on the tail parameters, and prove a quadratic lower bound on the number of non-isomorphic graphs for infinitely many values of n. These results provide a unified structural framework for central 2-trees with bounded tail degrees and highlight sharp transitions between rigidity and combinatorial growth.