Spectral Turán-type problems on sparse spanning graphs

Let F be a graph, and let EX ( n , F ) , SPEX A ( n , F ) , and SPEX Q ( n , F ) denote the classes of graphs that attain, respectively, the maximum number of edges, the maximum adjacency spectral radius, and the maximum signless Laplacian spectral radius over all n -vertex graphs that do not contain F as a subgraph. A fundamental problem in spectral extremal graph theory is to characterize all graphs F for which SPEX A ( n , F ) ⊆ EX ( n , F ) when n is sufficiently large. Establishing the conjecture of Cioabă et al. (2022) [10] , Wang et al. (2023) [54] proved that: for any graph F such that the graphs in EX ( n , F ) are Turán graphs plus O ( 1 ) edges, SPEX A ( n , F ) ⊆ EX ( n , F ) for sufficiently large n . In addition, another interesting problem in spectral extremal graph theory is to characterize all graphs F such that SPEX A ( n , F ) = SPEX Q ( n , F ) for sufficiently large n . In this paper, we give new contribution to the problems mentioned above. First, we present a substantial collection of examples of graphs F for which SPEX A ( n , F ) ⊆ EX ( n , F ) holds when n is sufficiently large, focusing on n -vertex graph F with no isolated vertices and maximum degree Δ ( F ) ≤ n / 40 . Second, under the same conditions on F , we prove that SPEX A ( n , F ) = SPEX Q ( n , F ) for sufficiently large n . These results can be seen as spectral analogs of a theorem by Alon and Yuster (2013) [1] . Furthermore, as immediate corollaries, we have tight spectral conditions for the existence of several classes of special graphs, including clique-factors, k -th power of Hamilton cycles and k -factors in graphs. The first special class of graphs gives a positive answer to a problem of Feng, and the second one extends a previous result of Yan et al.