Remarks on D -integral complete multipartite graphs

Pavel Híc; Milan Pokorný

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 2, page 457-464
  • ISSN: 0011-4642

Abstract

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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 < p 2 < p 3 , and K p 1 , p 2 , p 3 , p 4 with p 1 < p 2 < p 3 < p 4 , as well as the infinite classes of distance integral complete multipartite graphs K a 1 p 1 , a 2 p 2 , ... , a s p s with s = 5 , 6 .

How to cite

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Híc, Pavel, and Pokorný, Milan. "Remarks on $D$-integral complete multipartite graphs." Czechoslovak Mathematical Journal 66.2 (2016): 457-464. <http://eudml.org/doc/280088>.

@article{Híc2016,
abstract = {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 $p_\{1\}<p_\{2\}<p_\{3\}$, and $K_\{p_\{1\},p_\{2\},p_\{3\},p_\{4\}\}$ with $p_\{1\}<p_\{2\}<p_\{3\}<p_\{4\}$, 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 $s=5,6$.},
author = {Híc, Pavel, Pokorný, Milan},
journal = {Czechoslovak Mathematical Journal},
keywords = {distance spectrum; integral graph; distance integral graph; complete multipartite graph},
language = {eng},
number = {2},
pages = {457-464},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Remarks on $D$-integral complete multipartite graphs},
url = {http://eudml.org/doc/280088},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Híc, Pavel
AU - Pokorný, Milan
TI - Remarks on $D$-integral complete multipartite graphs
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 2
SP - 457
EP - 464
AB - 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 $p_{1}<p_{2}<p_{3}$, and $K_{p_{1},p_{2},p_{3},p_{4}}$ with $p_{1}<p_{2}<p_{3}<p_{4}$, 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 $s=5,6$.
LA - eng
KW - distance spectrum; integral graph; distance integral graph; complete multipartite graph
UR - http://eudml.org/doc/280088
ER -

References

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