Geodesic
Remarks on $D$-integral complete multipartite graphs
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 457-464
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

A graph is called distance integral (or $D$-integral) if all eigenvalues of its distance matrix are integers. In their study of $D$-integral complete multipartite graphs, Yang and Wang (2015) posed two questions on the existence of such graphs. We resolve these questions and present some further results on $D$-integral complete multipartite graphs. We give the first known distance integral complete multipartite graphs $K_{p_{1},p_{2},p_{3}}$ with $p_{1}
A graph is called distance integral (or $D$-integral) if all eigenvalues of its distance matrix are integers. In their study of $D$-integral complete multipartite graphs, Yang and Wang (2015) posed two questions on the existence of such graphs. We resolve these questions and present some further results on $D$-integral complete multipartite graphs. We give the first known distance integral complete multipartite graphs $K_{p_{1},p_{2},p_{3}}$ with $p_{1}$, and $K_{p_{1},p_{2},p_{3},p_{4}}$ with $p_{1}$, as well as the infinite classes of distance integral complete multipartite graphs $K_{a_{1} p_{1},a_{2} p_{2},\ldots ,a_{s} p_{s}}$ with $s=5,6$.
DOI : 10.1007/s10587-016-0268-8
Classification : 05C50
Keywords: distance spectrum; integral graph; distance integral graph; complete multipartite graph 
@article{10_1007_s10587_016_0268_8,
     author = {H{\'\i}c, Pavel and Pokorn\'y, Milan},
     title = {Remarks on $D$-integral complete multipartite graphs},
     journal = {Czechoslovak Mathematical Journal},
     pages = {457--464},
     year = {2016},
     volume = {66},
     number = {2},
     doi = {10.1007/s10587-016-0268-8},
     mrnumber = {3519614},
     zbl = {06604479},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0268-8/}
}
TY  - JOUR
AU  - Híc, Pavel
AU  - Pokorný, Milan
TI  - Remarks on $D$-integral complete multipartite graphs
JO  - Czechoslovak Mathematical Journal
PY  - 2016
SP  - 457
EP  - 464
VL  - 66
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0268-8/
DO  - 10.1007/s10587-016-0268-8
LA  - en
ID  - 10_1007_s10587_016_0268_8
ER  - 
%0 Journal Article
%A Híc, Pavel
%A Pokorný, Milan
%T Remarks on $D$-integral complete multipartite graphs
%J Czechoslovak Mathematical Journal
%D 2016
%P 457-464
%V 66
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0268-8/
%R 10.1007/s10587-016-0268-8
%G en
%F 10_1007_s10587_016_0268_8
Híc, Pavel; Pokorný, Milan. Remarks on $D$-integral complete multipartite graphs. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 457-464. doi: 10.1007/s10587-016-0268-8

[1] Andreescu, T., Andrica, D., Cucurezeanu, I.: An Introduction to Diophantine Equations. A Problem-Based Approach. Birkhäuser New York (2010). | MR | Zbl

[2] Aouchiche, M., Hansen, P.: Distance spectra of graphs: a survey. Linear Algebra Appl. 458 (2014), 301-386. | MR | Zbl

[3] Balińska, K., Cvetković, D., Radosavljević, Z., Simić, S., Stevanović, D.: A survey on integral graphs. Publ. Elektroteh. Fak., Univ. Beogr., Ser. Mat. 13 (2002), 42-65. | MR | Zbl

[4] Clark, J., Kettle, S. F. A.: Incidence and distance matrices. Inorg. Chim. Acta 14 (1975), 201-205. | DOI

[5] Du, Z., Ilić, A., Feng, L.: Further results on the distance spectral radius of graphs. Linear Multilinear Algebra 61 (2013), 1287-1301. | DOI | MR | Zbl

[6] Güngör, A. D., Bozkurt, Ş. B.: On the distance spectral radius and the distance energy of graphs. Linear Multilinear Algebra 59 (2011), 365-370. | DOI | MR | Zbl

[7] Harary, F., Schwenk, A. J.: Which graphs have integral spectra?. Graphs Combinatorics, Proc. Capital Conf., Washington, 1973, Lect. Notes Math. 406 Springer, Berlin (1974), 45-51. | DOI | MR

[8] Ilić, A.: Distance spectra and distance energy of integral circulant graphs. Linear Algebra Appl. 433 (2010), 1005-1014. | MR | Zbl

[9] Indulal, G., Gutman, I., Vijayakumar, A.: On distance energy of graphs. MATCH Commun. Math. Comput. Chem. 60 (2008), 461-472. | MR | Zbl

[10] Pokorný, M., Híc, P., Stevanović, D., Milošević, M.: On distance integral graphs. Discrete Math. 338 (2015), 1784-1792. | DOI | MR | Zbl

[11] Renteln, P.: The distance spectra of Cayley graphs of Coxeter groups. Discrete Math. 311 (2011), 738-755. | DOI | MR | Zbl

[12] Stevanović, D., Indulal, G.: The distance spectrum and energy of the compositions of regular graphs. Appl. Math. Lett. 22 (2009), 1136-1140. | DOI | MR | Zbl

[13] Stevanović, D., Milošević, M., Híc, P., Pokorný, M.: Proof of a conjecture on distance energy of complete multipartite graphs. MATCH Commun. Math. Comput. Chem. 70 (2013), 157-162. | MR | Zbl

[14] Yang, R., Wang, L.: Distance integral complete multipartite graphs with $s=5,6$. (2015), 6 pages Preprint arXiv:1511.04983v1 [math.CO]. | MR

[15] Yang, R., Wang, L.: Distance integral complete $r$-partite graphs. Filomat 29 (2015), 739-749. | DOI | MR

[16] Zhou, B., Ilić, A.: On distance spectral radius and distance energy of graphs. MATCH Commun. Math. Comput. Chem. 64 (2010), 261-280. | MR | Zbl

Cité par Sources :