Spectral extremal results with forbidding linear forests

Abstract

The Turán type extremal problem asks to maximize the number of edges over all graphs which do not contain fixed subgraphs. Similarly, the spectral Turán type extremal problem asks to maximize spectral radius of all graphs which do not contain fixed subgraphs. In this paper, we determine the maximum spectral radius of all graphs without containing a linear forest as a subgraph and characterize all corresponding extremal graphs. In addition, the maximum number of edges and spectral radius of all bipartite graphs without containing $k\cdot P_3$ as a subgraph are obtained and all extremal graphs are also characterized. Moreover, some relations between Tuán type extremal problems and spectral Turán type extremal problems are discussed.

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