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 $\delta(i)= \sum\limits_{\{i,j\} \in E(G)}d(j)$, 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 $n\times n$ 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.