Geodesic
Bounds for the (Laplacian) spectral radius of graphs with parameter $\alpha $
Czechoslovak Mathematical Journal, Tome 62 (2012) no. 2, pp. 567-580
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Let $G$ be a simple connected graph of order $n$ with degree sequence $(d_1,d_2,\ldots ,d_n)$. Denote $(^\alpha t)_i = \sum \nolimits _{j\colon i \sim j} {d_j^\alpha }$, $(^\alpha m)_i = {(^\alpha t)_i }/{d_i^\alpha }$ and $(^\alpha N)_i = \sum \nolimits _{j\colon i \sim j} {(^\alpha t)_j }$, where $\alpha $ is a real number. Denote by $\lambda _1(G)$ and $\mu _1(G)$ the spectral radius of the adjacency matrix and the Laplacian matrix of $G$, respectively. In this paper, we present some upper and lower bounds of $\lambda _1(G)$ and $\mu _1(G)$ in terms of $(^\alpha t)_i $, $(^\alpha m)_i $ and $(^\alpha N)_i $. Furthermore, we also characterize some extreme graphs which attain these upper bounds. These results theoretically improve and generalize some known results.
Let $G$ be a simple connected graph of order $n$ with degree sequence $(d_1,d_2,\ldots ,d_n)$. Denote $(^\alpha t)_i = \sum \nolimits _{j\colon i \sim j} {d_j^\alpha }$, $(^\alpha m)_i = {(^\alpha t)_i }/{d_i^\alpha }$ and $(^\alpha N)_i = \sum \nolimits _{j\colon i \sim j} {(^\alpha t)_j }$, where $\alpha $ is a real number. Denote by $\lambda _1(G)$ and $\mu _1(G)$ the spectral radius of the adjacency matrix and the Laplacian matrix of $G$, respectively. In this paper, we present some upper and lower bounds of $\lambda _1(G)$ and $\mu _1(G)$ in terms of $(^\alpha t)_i $, $(^\alpha m)_i $ and $(^\alpha N)_i $. Furthermore, we also characterize some extreme graphs which attain these upper bounds. These results theoretically improve and generalize some known results.
DOI : 10.1007/s10587-012-0030-9
Classification : 05C50, 15A18
Keywords: graph; adjacency matrix; Laplacian matrix; spectral radius; bound 
@article{10_1007_s10587_012_0030_9,
     author = {Tian, Gui-Xian and Huang, Ting-Zhu},
     title = {Bounds for the {(Laplacian)} spectral radius of graphs with parameter $\alpha $},
     journal = {Czechoslovak Mathematical Journal},
     pages = {567--580},
     year = {2012},
     volume = {62},
     number = {2},
     doi = {10.1007/s10587-012-0030-9},
     mrnumber = {2990195},
     zbl = {1265.05418},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-012-0030-9/}
}
TY  - JOUR
AU  - Tian, Gui-Xian
AU  - Huang, Ting-Zhu
TI  - Bounds for the (Laplacian) spectral radius of graphs with parameter $\alpha $
JO  - Czechoslovak Mathematical Journal
PY  - 2012
SP  - 567
EP  - 580
VL  - 62
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-012-0030-9/
DO  - 10.1007/s10587-012-0030-9
LA  - en
ID  - 10_1007_s10587_012_0030_9
ER  - 
%0 Journal Article
%A Tian, Gui-Xian
%A Huang, Ting-Zhu
%T Bounds for the (Laplacian) spectral radius of graphs with parameter $\alpha $
%J Czechoslovak Mathematical Journal
%D 2012
%P 567-580
%V 62
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-012-0030-9/
%R 10.1007/s10587-012-0030-9
%G en
%F 10_1007_s10587_012_0030_9
Tian, Gui-Xian; Huang, Ting-Zhu. Bounds for the (Laplacian) spectral radius of graphs with parameter $\alpha $. Czechoslovak Mathematical Journal, Tome 62 (2012) no. 2, pp. 567-580. doi: 10.1007/s10587-012-0030-9

[1] Berman, A., Zhang, X.-D.: On the spectral radius of graphs with cut vertices. J. Combin. Theory, Ser. B 83 (2001), 233-240. | DOI | MR | Zbl

[2] Brankov, V., Hansen, P., Stevanović, D.: Automated cunjectures on upper bounds for the largest Laplacian eigenvalue of graphs. Linear Algebra Appl. 414 (2006), 407-424. | MR

[3] Cvetković, D., Doob, M., Sachs, H.: Spectra of Graphs. Theory and Application. Deutscher Verlag der Wissenschaften Berlin (1980). | MR | Zbl

[4] Das, K. C., Kumar, P.: Some new bounds on the spectral radius of graphs. Discrete Math. 281 (2004), 149-161. | DOI | MR | Zbl

[5] Favaron, O., Mahéo, M., Saclé, J.-F.: Some eigenvalue properties in graphs (Conjectures of Graffiti---II). Discrete Math. 111 (1993), 197-220. | DOI | MR | Zbl

[6] Hofmeister, M.: Spectral radius and degree sequence. Math. Nachr. 139 (1988), 37-44. | DOI | MR | Zbl

[7] Hong, Y., Zhang, X.-D.: Sharp upper and lower bounds for largest eigenvalue of the Laplacian matrices of trees. Discrete Math. 296 (2005), 187-197. | DOI | MR | Zbl

[8] Liu, H., Lu, M.: Bounds for the Laplacian spectral radius of graphs. Linear Multilinear Algebra 58 (2010), 113-119. | DOI | MR | Zbl

[9] Liu, H., Lu, M., Tian, F.: Some upper bounds for the energy of graphs. J. Math. Chem. 41 (2007), 45-57. | DOI | MR | Zbl

[10] Nikiforov, V.: The energy of graphs and matrices. J. Math. Anal. Appl. 326 (2007), 1472-1475. | DOI | MR | Zbl

[11] Shi, L.: Bounds on the (Laplacian) spectral radius of graphs. Linear Algebra Appl. 422 (2007), 755-770. | DOI | MR | Zbl

[12] Tian, G.-X., Huang, T.-Z., Zhou, B.: A note on sum of powers of the Laplacian eigenvalues of bipartite graphs. Linear Algebra Appl. 430 (2009), 2503-2510. | MR | Zbl

[13] Yu, A., Lu, M., Tian, F.: On the spectral radius of graphs. Linear Algebra Appl. 387 (2004), 41-49. | DOI | MR | Zbl

[14] Yu, A., Lu, M., Tian, F.: New upper bounds for the energy of graphs. MATCH Commun. Math. Comput. Chem. 53 (2005), 441-448. | MR | Zbl

[15] Zhou, B.: Energy of a graph. MATCH Commun. Math. Comput. Chem. 51 (2004), 111-118. | MR | Zbl

Cité par Sources :