The minimum $\varepsilon$-spectral radius of $t$-clique trees with given diameter

The eccentricity matrix $\varepsilon(G)$ of a graph $G$ is defined as \begin{equation}\varepsilon(G)_{uv}= \begin{cases}d_{uv} & d_{uv}=min\{e(u),e(v)\},\\0 & d_{uv} < min\{e(u),e(v)\}. \notag\end{cases}\end{equation} Let $T_t$ be a $t$-clique tree corresponding to the tree $T($underlying graph of $T_t)$ with order $n'=(n-1)t+1$ and diameter $d$. In this paper, we identify the extremal $t$-clique trees with given diameter having the minimum $\varepsilon$-spectral radius. Simultaneously, we calculate the lower bound of $\varepsilon$-spectral radius of $t$-clique trees when $n-d$ is odd.