Spectral Extrema of Graphs With Fixed Size: Forbidden a Fan Graph, a Friendship Graph, or a Theta Graph

It is well‐known that Brualdi‐Hoffman‐Turán‐type problem inquiries about the maximum spectral radius of an ‐free graph with edges. This can be regarded as a spectral characterization of the existence of the subgraph within . A significant contribution to this problem was made by Nikiforov (2002). He proved that for every ‐free graph of size . Let be the theta graph, which is obtained by connecting two vertices with three internally disjoint paths of lengths , respectively. Let be the fan graph, that is, the join of a and a path , and let be the friendship graph obtained from triangles by sharing a common vertex. In this paper, we utilize the ‐core method and spectral techniques to address some spectral extrema of graphs with fixed size. Firstly, we show that, for with , if is ‐free, then . Equality holds if and only if . This confirms a conjecture of Yu et al. and improves a recent result of Li et al. Secondly, we show that, for with , if is ‐free of size , then . Equality holds if and only if . This confirms a conjecture proposed by Li et al. Finally, we identify the ‐free graph of size having the largest spectral radius, where and . A further research problem is also proposed.