Sharp spectral Moon--Moser-type theorems in the linear range via feasible graph parameters

Moon and Moser proved a sharp edge-extremal theorem for Hamilton cycles in balanced bipartite graphs with minimum degree at least $k$.

Li and Ning obtained spectral analogues for Hamiltonicity in balanced bipartite graphs of order $2n$ and for traceability in nearly balanced bipartite graphs with part sizes $n$ and $n-1$, under the assumption $n\ge (k+1)^2$.

We show that their sharp spectral thresholds remain valid in the linear ranges $n\ge 2k$ and $n\ge 2k+1$, respectively.

More precisely, we determine the extremal values of the adjacency spectral radius and the signless Laplacian spectral radius for non-Hamiltonian balanced bipartite graphs with minimum degree $\delta(G)\ge k$, and for non-traceable nearly balanced bipartite graphs with $\delta(G)\ge k$.

In each case, the extremal graph is unique up to isomorphism.

Our proof is based on feasible graph parameters: parameters that increase under edge addition and are nondecreasing under Kelmans operations.

This yields Moon--Moser type extremal theorems for a general class of parameters, from which the spectral results follow.