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DIAMETER OF COMMUTING GRAPHS OF SYMPLECTIC ALGEBRAS

Part of: Graph theory Basic linear algebra

Published online by Cambridge University Press:  04 June 2019

XIANYA GENG Open the ORCID record for XIANYA GENG [Opens in a new window] ,
LITING FAN Open the ORCID record for LITING FAN [Opens in a new window]  and
XIAOBIN MA Open the ORCID record for XIAOBIN MA [Opens in a new window]
XIANYA GENG*
Affiliation:
School of Mathematics and Big Data, Anhui University of Science and Technology, Huainan, PR China email gengxianya@sina.com
LITING FAN
Affiliation:
School of Mathematics and Big Data, Anhui University of Science and Technology, Huainan, PR China email 1275371706@qq.com
XIAOBIN MA
Affiliation:
School of Mathematics and Big Data, Anhui University of Science and Technology, Huainan, PR China email xiaobintiandi@163.com
*
✉gengxianya@sina.com
Rights & Permissions [Opens in a new window]

Abstract

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Let $F$ be an algebraically closed field of characteristic $0$ and let $\operatorname{sp}(2l,F)$ be the rank $l$ symplectic algebra of all $2l\times 2l$ matrices $x=\big(\!\begin{smallmatrix}A & B\\ C & -A^{t}\end{smallmatrix}\!\big)$ over $F$, where $A^{t}$ is the transpose of $A$ and $B,C$ are symmetric matrices of order $l$. The commuting graph $\unicode[STIX]{x1D6E4}(\operatorname{sp}(2l,F))$ of $\operatorname{sp}(2l,F)$ is a graph whose vertex set consists of all nonzero elements in $\operatorname{sp}(2l,F)$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy=yx$. We prove that the diameter of $\unicode[STIX]{x1D6E4}(\operatorname{sp}(2l,F))$ is $4$ when $l>2$.

Keywords

commuting graphsdiameterLie algebras

MSC classification

Primary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)
Secondary: 15A27: Commutativity 15A30: Algebraic systems of matrices
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 100 , Issue 3 , December 2019 , pp. 419 - 427
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

Footnotes

Financially supported by the National Natural Science Foundation of China (Grant Nos. 11671164, 11271149).

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