| Title:
|
On the spectral radius of $\ddag $-shape trees (English) |
| Author:
|
Ma, Xiaoling |
| Author:
|
Wen, Fei |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
63 |
| Issue:
|
3 |
| Year:
|
2013 |
| Pages:
|
777-782 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $A(G)$ be the adjacency matrix of $G$. The characteristic polynomial of the adjacency matrix $A$ is called the characteristic polynomial of the graph $G$ and is denoted by $\phi (G, \lambda )$ or simply $\phi (G)$. The spectrum of $G$ consists of the roots (together with their multiplicities) $\lambda _1(G)\geq \lambda _2(G)\geq \ldots \geq \lambda _n(G)$ of the equation $\phi (G, \lambda )=0$. The largest root $\lambda _1(G)$ is referred to as the spectral radius of $G$. A $\ddag $-shape is a tree with exactly two of its vertices having maximal degree 4. We will denote by $G(l_1, l_2, \ldots , l_7)$ $(l_1\geq 0$, $l_i\geq 1$, $i=2,3,\ldots , 7)$ a $\ddag $-shape tree such that $G(l_1, l_2, \ldots , l_7)-u-v=P_{l_1}\cup P_{l_2}\cup \ldots \cup P_{l_7}$, where $u$ and $v$ are the vertices of degree 4. In this paper we prove that $3\sqrt {2}/{2}< \lambda _1(G(l_1, l_2, \ldots , l_7))< {5}/{2}$. (English) |
| Keyword:
|
spectra of graphs |
| Keyword:
|
spectral radius |
| Keyword:
|
$\ddag $-shape tree |
| MSC:
|
05C50 |
| idZBL:
|
Zbl 06282109 |
| idMR:
|
MR3125653 |
| DOI:
|
10.1007/s10587-013-0051-z |
| . |
| Date available:
|
2013-10-07T12:07:39Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143488 |
| . |
| Reference:
|
[1] Cvetiović, D., Doob, M., Sachs, H.: Spectra of Graphs. Theory and Applications.VEB Deutscher Verlag der Wissenschaften, Berlin (1980). |
| Reference:
|
[2] Harary, F.: Graph Theory.Addison-Wesley Series in Mathematics. Addison-Wesley Publishing Company. IX, Reading, Mass.-Menlo Park London (1969). Zbl 0196.27202, MR 0256911 |
| Reference:
|
[3] Hoffman, A. J., Smith, J. H.: On the spectral radii of topologically equivalent graphs.Recent Adv. Graph Theory, Proc. Symp. Prague 1974 Academia, Praha, 1975 273-281. Zbl 0327.05125, MR 0404028 |
| Reference:
|
[4] Hu, S. B.: On the spectral radius of $H$-shape trees.Int. J. Comput. Math. 87 (2010), 976-979. Zbl 1209.05047, MR 2665706, 10.1080/00207160802051022 |
| Reference:
|
[5] Godsil, C. D.: Algebraic Combinatorics.Chapman and Hall, New York (1993). Zbl 0784.05001, MR 1220704 |
| Reference:
|
[6] Godsil, C. D.: Spectra of trees.Convexity and Graph Theory. Proc. Conf., Israel 1981, Ann. Discrete Math. 20 (1984), 151-159. Zbl 0559.05040, MR 0791025 |
| Reference:
|
[7] 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:
|
[8] Woo, R., Neumaier, A.: On graphs whose spectral radius is bounded by $\frac32\sqrt{2}$.Graphs Comb. 23 (2007), 713-726. MR 2365422, 10.1007/s00373-007-0745-9 |
| . |
