| Title:
|
On the signless Laplacian spectral characterization of the line graphs of $T$-shape trees (English) |
| Author:
|
Wang, Guoping |
| Author:
|
Guo, Guangquan |
| Author:
|
Min, Li |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
64 |
| Issue:
|
2 |
| Year:
|
2014 |
| Pages:
|
311-325 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A graph is determined by its signless Laplacian spectrum if no other non-isomorphic graph has the same signless Laplacian spectrum (simply $G$ is $DQS$). Let $T(a,b,c)$ denote the $T$-shape tree obtained by identifying the end vertices of three paths $P_{a+2}$, $P_{b+2}$ and $P_{c+2}$. We prove that its all line graphs $\mathcal {L}(T(a,b,c))$ except $\mathcal {L}(T(t,t,2t+1))$ ($t\geq 1$) are $DQS$, and determine the graphs which have the same signless Laplacian spectrum as $\mathcal {L}(T(t,t,2t+1))$. Let $\mu _1(G)$ be the maximum signless Laplacian eigenvalue of the graph $G$. We give the limit of $\mu _1(\mathcal {L}(T(a,b,c)))$, too. (English) |
| Keyword:
|
signless Laplacian spectrum |
| Keyword:
|
cospectral graphs |
| Keyword:
|
$T$-shape tree |
| MSC:
|
05C50 |
| MSC:
|
15A18 |
| idZBL:
|
Zbl 06391496 |
| idMR:
|
MR3277738 |
| DOI:
|
10.1007/s10587-014-0103-z |
| . |
| Date available:
|
2014-11-10T09:29:08Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144000 |
| . |
| Reference:
|
[1] Cvetković, D. M., Doob, M., Sachs, H.: Spectra of Graphs. Theory and Applications.3rd rev. a. enl. J. A. Barth, Leipzig (1995). Zbl 0824.05046, MR 1324340 |
| Reference:
|
[2] Cvetković, D., Rowlinson, P., Simić, S. K.: Signless Laplacians of finite graphs.Linear Algebra Appl. 423 (2007), 155-171. MR 2312332, 10.1016/j.laa.2007.01.009 |
| Reference:
|
[3] Cvetković, D., Rowlinson, P., Simić, S. K.: Eigenvalue bounds for the signless Laplacian.Publ. Inst. Math., Nouv. Sér. 81 (2007), 11-27. Zbl 1164.05038, MR 2401311, 10.2298/PIM0795011C |
| Reference:
|
[4] Cvetković, D., Simić, S. K.: Towards a spectral theory of graphs based on signless Laplacian. I.Publ. Inst. Math., Nouv. Sér. 85 (2009), 19-33. MR 2536686 |
| Reference:
|
[5] Cvetković, D., Simić, S. K.: Towards a spectral theory of graphs based on signless Laplacian. II.Linear Algebra Appl. 432 (2010), 2257-2272. MR 2599858, 10.1016/j.laa.2009.05.020 |
| Reference:
|
[6] Ghareghani, N., Omidi, G. R., Tayfeh-Rezaie, B.: Spectral characterization of graphs with index at most $\sqrt{2+\sqrt{5}}$.Linear Algebra Appl. 420 (2007), 483-489. Zbl 1107.05058, MR 2278224 |
| Reference:
|
[7] Omidi, G. R.: On a signless Laplacian spectral characterizaiton of $T$-shape trees.Linear Algebra Appl. 431 (2009), 1607-1615. MR 2555062, 10.1016/j.laa.2009.05.035 |
| Reference:
|
[8] Ramezani, F., Broojerdian, N., Tayfeh-Rezaie, B.: A note on the spectral characterization of $\theta$-graphs.Linear Algebra Appl. 431 (2009), 626-632. Zbl 1203.05098, MR 2535538 |
| Reference:
|
[9] Dam, E. R. van, Haemers, W. H.: Which graphs are determined by their spectrum?.Linear Algebra Appl. Special issue on the Combinatorial Matrix Theory Conference (Pohang, 2002) 373 (2003), 241-272. MR 2022290 |
| Reference:
|
[10] Dam, E. R. van, Haemers, W. H.: Developments on spectral characterizations of graphs.Discrete Math. 309 (2009), 576-586. MR 2499010, 10.1016/j.disc.2008.08.019 |
| Reference:
|
[11] Wang, J. F., Huang, Q. X., Belardo, F., Marzi, E. M. L.: On the spectral characterizations of $\infty$-graphs.Discrete Math. 310 (2010), 1845-1855. Zbl 1231.05174, MR 2629903, 10.1016/j.disc.2010.01.021 |
| Reference:
|
[12] Wang, W., Xu, C. X.: On the spectral characterization of $T$-shape trees.Linear Algebra Appl. 414 (2006), 492-501. Zbl 1086.05050, MR 2214401, 10.1016/j.laa.2005.10.031 |
| Reference:
|
[13] Wang, W., Xu, C. X.: Note: The $T$-shape tree is determined by its Laplacian spectrum.Linear Algebra Appl. 419 (2006), 78-81. MR 2263111 |
| Reference:
|
[14] Zhang, Y. P., Liu, X. G., Zhang, B. Y., Yong, X. R.: The lollipop graph is determined by its $Q$-spectrum.Discrete Math. 309 (2009), 3364-3369. Zbl 1182.05084, MR 2526754, 10.1016/j.disc.2008.09.052 |
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