1 results
GRAPHS DETERMINED BY THEIR
$T$-GAIN SPECTRA
- Part of
-
- Journal:
- Bulletin of the Australian Mathematical Society / Volume 103 / Issue 2 / April 2021
- Published online by Cambridge University Press:
- 03 July 2020, pp. 195-203
- Print publication:
- April 2021
-
- Article
- Export citation
-
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.
