Abstract
The complete split graph, denoted by Sn,k, is the graph obtained by joining a copy of Kk to n−k isolated vertices. As a special threshold graph, it plays an important role in graph theory and spectral graph theory. Some known results tell us that if q(G)>q(Sn,k) then G can contain many kinds of subgraphs, such as paths, linear forests, friendship graphs, flowers, and so on (where q(G) is the Q-index of G). Motivated by these results, we are interested in finding larger subgraphs in G. In this paper, we show that if q(G)>q(Sn,k) then G contains a fan H2k, where H2k denotes the graph obtained by joining a vertex to a path of order 2k. This implies several previous results, and a conjecture proposed by Li and Peng.
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