FIRST ZAGREB MATRIX AND ENERGY OF A $T_2$ HYPERGRAPH

Let $H$ be a $T_2$ hypergraph of order $n\geq4$. The first Zagreb matrix of $H$, denoted by $Z(H)$ is defined as the square matrix of order $n$, whose $(i,j)^{th}$ entry is $d_i+d_j$ if $x_i$ and $x_j$ are adjacent and zero for other cases. The first Zagreb energy $ZE(H)$ of $H$ is the sum of the absolute values of the eigenvalues of $Z(H)$. It is shown that, for a $T_2$ hypergraph $ZE(H)\leq\left\lceil\frac{\sqrt{2}(n^2+3n+1)}{\sqrt{3}}\right\rceil.$