Extremal and spectral extremal graphs for powers of cycles when the cycle length is divisible by the power parameter

Characterize, for integers k,p≥2 and m=p(k+1), all sufficiently large graphs in the extremal family EX(n,{C_m^k}) and the spectral extremal family SPEX(n,{C_m^k}), and prove that each such graph has the form H∨T(n−|H|,k−1), where H is a C_{2p}-free graph.

Background

The paper studies extremal graphs and spectral extremal graphs avoiding specified subgraphs, with particular emphasis on powers C_m^k of cycles. Theorem 1.7 settles the case m=p(k+1)+h with 1≤h≤k, proving that a specific join of a clique and a Turán graph is uniquely extremal both in edge count and spectral radius for sufficiently large n.

The remaining case m=p(k+1), where the cycle length is exactly divisible by k+1, is structurally different and is not resolved by Theorem 1.7. The authors conjecture that every extremal and spectral extremal graph in this case is obtained by joining a C_{2p}-free graph H to a Turán graph with k−1 parts on the remaining vertices. Here C_m^k denotes the k-th power of the cycle C_m, EX(n,{C_m^k}) denotes the C_m^k-free graphs of order n with the maximum number of edges, and SPEX(n,{C_m^k}) denotes the C_m^k-free graphs of order n with the maximum spectral radius.