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Graphs and Combinatorics

, Volume 35, Issue 1, pp 335–351 | Cite as

Spectral Extremal Results with Forbidding Linear Forests

  • Ming-Zhu Chen
  • A-Ming Liu
  • Xiao-Dong ZhangEmail author
Original Paper
  • 85 Downloads

Abstract

The Turán type extremal problems ask to maximize the number of edges over all graphs which do not contain fixed subgraphs. Similarly, their spectral counterparts ask 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 a linear forest as a subgraph and all the extremal graphs. In addition, the maximum number of edges and spectral radius of all bipartite graphs without \(k\cdot P_3\) as a subgraph are obtained and all the extremal graphs are also determined. Moreover, some relations between Turán type extremal problems and their spectral counterparts are discussed.

Keywords

Turán type extremal problem Spectral counterparts Linear forest Spectral radius Bipartite graph 

Mathematics Subject Classification

05C50 05C35 

Notes

Acknowledgements

The authors would like to thank the anonymous referee for many helpful and constructive suggestions to an earlier version of this paper, which results in an great improvement.

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Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical Sciences, MOE-LSC, SHL-MACShanghai Jiao Tong UniversityShanghaiPeople’s Republic of China

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