# The Least Eigenvalue of Graphs whose Complements Are Uni- cyclic

• Volume: 35, Issue: 2, page 249-260
• ISSN: 2083-5892

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## Abstract

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A graph in a certain graph class is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum among all graphs in that class. Bell et al. have identified a subclass within the connected graphs of order n and size m in which minimizing graphs belong (the complements of such graphs are either disconnected or contain a clique of size n/2 ). In this paper we discuss the minimizing graphs of a special class of graphs of order n whose complements are connected and contains exactly one cycle (namely the class Ucn of graphs whose complements are unicyclic), and characterize the unique minimizing graph in Ucn when n ≥ 20.

## How to cite

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Yi Wang, et al. "The Least Eigenvalue of Graphs whose Complements Are Uni- cyclic." Discussiones Mathematicae Graph Theory 35.2 (2015): 249-260. <http://eudml.org/doc/271086>.

@article{YiWang2015,
abstract = {A graph in a certain graph class is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum among all graphs in that class. Bell et al. have identified a subclass within the connected graphs of order n and size m in which minimizing graphs belong (the complements of such graphs are either disconnected or contain a clique of size n/2 ). In this paper we discuss the minimizing graphs of a special class of graphs of order n whose complements are connected and contains exactly one cycle (namely the class Ucn of graphs whose complements are unicyclic), and characterize the unique minimizing graph in Ucn when n ≥ 20.},
author = {Yi Wang, Yi-Zheng Fan, Xiao-Xin Li, Fei-Fei Zhang},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {unicyclic graph; complement; adjacency matrix; least eigen- value; least eigenvalue},
language = {eng},
number = {2},
pages = {249-260},
title = {The Least Eigenvalue of Graphs whose Complements Are Uni- cyclic},
url = {http://eudml.org/doc/271086},
volume = {35},
year = {2015},
}

TY - JOUR
AU - Yi Wang
AU - Yi-Zheng Fan
AU - Xiao-Xin Li
AU - Fei-Fei Zhang
TI - The Least Eigenvalue of Graphs whose Complements Are Uni- cyclic
JO - Discussiones Mathematicae Graph Theory
PY - 2015
VL - 35
IS - 2
SP - 249
EP - 260
AB - A graph in a certain graph class is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum among all graphs in that class. Bell et al. have identified a subclass within the connected graphs of order n and size m in which minimizing graphs belong (the complements of such graphs are either disconnected or contain a clique of size n/2 ). In this paper we discuss the minimizing graphs of a special class of graphs of order n whose complements are connected and contains exactly one cycle (namely the class Ucn of graphs whose complements are unicyclic), and characterize the unique minimizing graph in Ucn when n ≥ 20.
LA - eng
KW - unicyclic graph; complement; adjacency matrix; least eigen- value; least eigenvalue
UR - http://eudml.org/doc/271086
ER -

## References

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1. [1] F.K. Bell, D. Cvetković, P. Rowlinson and S. Simić, Graph for which the least eigenvalues is minimal, I , Linear Algebra Appl. 429 (2008) 234-241. doi:10.1016/j.laa.2008.02.032[Crossref] Zbl1149.05030
2. [2] F.K. Bell, D. Cvetković, P. Rowlinson and S. Simić, Graph for which the least eigenvalues is minimal, II , Linear Algebra Appl. 429 (2008) 2168-2179. doi:10.1016/j.laa.2008.06.018[Crossref] Zbl1144.05313
3. [3] D. Cvetković and P. Rowlinson, The largest eigenvalues of a graph: a survey, Linear Multilinear Algebra 28 (1990) 3-33. doi:10.1080/03081089008818026[Crossref] Zbl0744.05031
4. [4] D. Cvetković, P. Rowlinson and S. Simić, Spectral Generalizations of Line Graphs: on Graph with Least Eigenvalue −2 (London Math. Soc., LNS 314, Cambridge Univ. Press, 2004). Zbl1061.05057
5. [5] Y.-Z. Fan, Y. Wang and Y.-B. Gao, Minimizing the least eigenvalues of unicyclic graphs with application to spectral spread, Linear Algebra Appl. 429 (2008) 577-588. doi:10.1016/j.laa.2008.03.012[WoS][Crossref] Zbl1143.05053
6. [6] Y.-Z. Fan, F.-F Zhang and Y. Wang, The least eigenvalue of the complements of trees, Linear Algebra Appl. 435 (2011) 2150-2155. doi:10.1016/j.laa.2011.04.011[WoS][Crossref]
7. [7] Y. Hong and J. Shu, Sharp lower bounds of the least eigenvalue of planar graphs, Linear Algebra Appl. 296 (1999) 227-232. doi:10.1016/S0024-3795(99)00129-9[Crossref]
8. [8] R. Liu, M. Zhai and J. Shu, The least eigenvalues of unicyclic graphs with n vertices and k pendant vertices, Linear Algebra Appl. 431 (2009) 657-665. doi:10.1016/j.laa.2009.03.016[WoS][Crossref] Zbl1169.05349
9. [9] M. Petrović, B. Borovićanin and T. Aleksić, Bicyclic graphs for which the least eigenvalue is minimum, Linear Algebra Appl. 430 (2009) 1328-1335. doi:10.1016/j.laa.2008.10.026[WoS][Crossref] Zbl1194.05093
10. [10] M. Petrović, T. Aleksić and S. Simić, Further results on the least eigenvalue of connected graphs, Linear Algebra Appl. 435 (2011) 2303-2313. doi:10.1016/j.laa.2011.04.030[WoS][Crossref] Zbl1222.05174
11. [11] Y.-Y. Tan and Y.-Z. Fan, The vertex (edge) independence number, vertex (edge) cover number and the least eigenvalue of a graph, Linear Algebra Appl. 433 (2010) 790-795. doi:10.1016/j.laa.2010.04.009[Crossref][WoS] Zbl1214.05085
12. [12] Y. Wang, Y. Qiao and Y.-Z. Fan, On the least eigenvalue of graphs with cut vertices, J. Math. Res. Exposition 30 (2010) 951-956. Zbl1240.05203
13. [13] Y. Wang and Y.-Z. Fan, The least eigenvalue of a graph with cut vertices, Linear Algebra Appl. 433 (2010) 19-27. doi:10.1016/j.laa.2010.01.030[WoS][Crossref]
14. [14] M.-L. Ye, Y.-Z. Fan and D. Liang, The least eigenvalue of graphs with given con- nectivity, Linear Algebra Appl. 430 (2009) 1375-1379. doi:10.1016/j.laa.2008.10.031 [Crossref]

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