Spectral extremal results with forbidding linear forests

TL;DR

This paper determines the maximum spectral radius of graphs avoiding linear forests, characterizes extremal graphs, and explores relations between Turán and spectral Turán extremal problems.

Contribution

It provides exact spectral radius bounds for graphs excluding linear forests and characterizes all extremal graphs, extending Turán type extremal results to spectral settings.

Findings

Maximum spectral radius for graphs without linear forests

Characterization of extremal graphs in these cases

Results on bipartite graphs avoiding multiple P3 subgraphs

Abstract

The Tur\'an type extremal problem asks to maximize the number of edges over all graphs which do not contain fixed subgraphs. Similarly, the spectral Tur\'an 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 Tu\'an type extremal problems and spectral Tur\'an type extremal problems are discussed.