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GRAPHS DETERMINED BY THEIR $T$-GAIN SPECTRA

Part of: Graph theory

Published online by Cambridge University Press:  03 July 2020

SAI WANG Open the ORCID record for SAI WANG [Opens in a new window] ,
DEIN WONG  and
FENGLEI TIAN
SAI WANG*
Affiliation:
School of Mathematics,China University of Mining and Technology, Xuzhou221116, Jiangsu, PR China email [email protected] Xuhai College, China University of Mining and Technology, Jiangsu, PR China
DEIN WONG
Affiliation:
School of Mathematics,China University of Mining and Technology, Xuzhou, PR China email [email protected]
FENGLEI TIAN
Affiliation:
School of Management,Qufu Normal University, Rizhao, 276826, Shandong, PR China email [email protected]
*
✉[email protected]
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Abstract

An undirected graph $G$ is determined by its $T$-gain spectrum (DTS) if every $T$-gain graph cospectral to $G$ is switching equivalent to $G$. We show that the complete graph $K_{n}$ and the graph $K_{n}-e$ obtained by deleting an edge from $K_{n}$ are DTS, the star $K_{1,n}$ is DTS if and only if $n\leq 2$, and an odd path $P_{2m+1}$ is not DTS if $m\geq 2$. We give an operation for constructing cospectral $T$-gain graphs and apply it to show that a tree of arbitrary order (at least $5$) is not DTS.

Keywords

complex unit gain graphsgraphs determined by their spectracospectral graphs

MSC classification

Primary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)
Secondary: 05C22: Signed and weighted graphs
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 2 , April 2021 , pp. 195 - 203
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

The first author is supported by the National Natural Science Foundation of China (No.11971474). The third author is supported by the Natural Science Foundation of Shandong Province (No. ZR2019BA016).

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