on spectral properties of neighbourhood second zagreb matrix of graphs

Let $g$ be a simple graph with vertex set $v(g)={1,2,dots,n}$ and $delta(i)= sumlimits_{{i,j} in e(g)}d(j)$, where $d(j)$ is the degree of the vertex $j$ in $g$. inspired by the second zagreb matrix and neighborhood first zagreb matrix of a graph, we introduce the neighborhood second zagreb matrix of $g$, denoted by $n_f(g)$. it is the $ntimes n$ matrix whose $ij$-th entry is equal to $delta(i)delta(j)$, if $i$ and $j$ are adjacent in $g$ and $0$, otherwise. the neighborhood second zagreb spectral radius $rho_{n_f}(g)$ is the largest eigenvalue of $n_f(g)$. the neighborhood second zagreb energy $mathcal{e}(n_f)$ of the graph $g$ is the sum of the absolute values of the eigenvalues of $n_f(g)$. in this paper, we obtain some spectral properties of $n_f(g)$. we provide sharp bounds for $rho_{n_f}(g)$ and $mathcal{e}(n_f)$, and obtain the corresponding extremal graphs.