X $ such that $ \rho(H_{k}(X)) =\Delta $.

Cospectral $ \mathbb{T} $-gain graph {#co-spec}
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For any simple graph $ G $, the collection of all $ \mathbb{T} $-gain graphs associated with $ G $ is denoted by $\mathcal{T}_{G}$. Let us consider $ \mathbb{T}_{G}:=\{A(\Phi): \Phi \in \mathcal{T}_{G} \} $. The spectrum of $ A(\Phi) $ is called the *$\mathbb{T}$-spectrum* of $ \Phi $, and the spectral radius of $A(\Phi)$ is called $\mathbb{T}$-spectral radius of $\Phi$. Let $ \Phi_{1}=(G, \varphi_{1})$ and $ \Phi_{2}=(G, \varphi_{2}) $ be two $ \mathbb{T}$-gain graphs having the same $ \mathbb{T}$-spectrum. Then $ \Phi_{1} $ and $ \Phi_{2} $ are called *$ \mathbb{T}$-cospectral*.

A considerable amount of literature is available on the construction of cospectral simple graphs. From the $ \mathbb{T}$-gain point of view, we may ask the same question about the class of $ \mathbb{T} $-cospectral gain graphs on $ G $.

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Let us first consider the example given in Figure \[fig1\]. From the figure, we have $ \sigma (A(\Phi_{1}))=\{-1.9318, -0.5176, 0.5176, 1.9318 \} $ and $ \sigma (A(\Phi_{2}))   =\{ 1.4811, -1, 0.3111, 2.1701\} $. Thus, in spite of having the same underlying graph $ G $, $ A(\Phi_{1}) $ and $A(\Phi_{2}) $ have different 