Skip to main content
SpringerLink
Log in

2-Walk linear graphs with small number of cycles

  • Published:
Wuhan University Journal of Natural Sciences

Abstract

A graph has exactly two main eigenvalues if and only if it is a 2-walk linear graph. In this paper, we show some necessary conditions that a 2-walk (a, b)-linear graph must obey. Using these conditions and some basic theorems in graph theory, we characterize all 2-walk linear graphs with small cyclic graphs without pendants. The results are given in sort on unicyclic, bicyclic, tricyclic graphs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bondy J A, Murty U S R. Graph Theory with Applications [M]. New York: Macmillan, 1976.

    Google Scholar 

  2. Borovićanin B, Grünewald S, Gutman I, et al. Harmonic graphs with small number of cycles[J]. Discrete Math, 2003, 265: 31–44.

    Article  MATH  MathSciNet  Google Scholar 

  3. Dress A, Gutman I. On the number of walks in a graph [J]. Appl Math Lett, 2003, 16: 797–801.

    Article  MATH  MathSciNet  Google Scholar 

  4. Hagos E M. Some results on graph spectra[J]. Linear Algebra Appl, 2002, 356: 103–111.

    Article  MATH  MathSciNet  Google Scholar 

  5. Dress A, Grünewald S, Stevanović D. Semiharmonic graphs with fixed cyclomatic number[J]. Appl Math Lett, 2004, 17(6): 623–629.

    Article  MATH  MathSciNet  Google Scholar 

  6. Dress A, Grünewald S, Stevanović D. Semiharmonic bicyclic graphs[J]. Appl Math Lett, 2005, 18(11): 1128–1138.

    Google Scholar 

  7. Grünewald S. Harmonic trees[J]. Appl Math Lett, 2002, 15(8): 1001–1004.

    Article  MATH  MathSciNet  Google Scholar 

  8. Hou Y, Zhou H. Trees with exactly two main eigenvalues[J]. Acta of Hunan Normal University, 2005, 28(2): 1–3(Ch).

    MATH  MathSciNet  Google Scholar 

  9. Hou Y, Tian F. Unicyclic graphs with exactly two main eigenvalues[ J]. Appl Math Lett, 2006, 19: 1143–1147.

    Article  MATH  MathSciNet  Google Scholar 

  10. Hu Ziquan, Li Shuchao, Zhu Chunfeng. Bicylic graphs with exactly two main eigenvalues [J]. Linear Algebra and Its Applications, 2009, 431(10): 1848–1857.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, Hubei, China

    Qiong Fan & Huan Qi

  2. School of Mathematics and Statistics, Huazhong Normal University, Wuhan, 430079, Hubei, China

    Qiong Fan

Authors
  1. Qiong Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huan Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Huan Qi.

Additional information

Foundation item: Supported by the National Natural Science Foundation of China (10671081)

Biography: FAN Qiong, female, Ph. D. c ndidate, Lecturer of Huazhong Normal University, research direction: graph theory.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fan, Q., Qi, H. 2-Walk linear graphs with small number of cycles. Wuhan Univ. J. Nat. Sci. 15, 375–379 (2010). https://doi.org/10.1007/s11859-010-0669-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11859-010-0669-8

Key words

CLC number

Search

Navigation