Reza Sharafdini

Let $G$ be a graph with the vertex set $V(G)=\{v_1,\ldots,v_n\}$. Due to [C. Godsil and G. Royele, \textit{Algebraic Graph Theory}, Springer, New York (2001)], a symmetric matrix $\mathcal{L}$ of order $n$ is called a \emph{generalized Laplacian} of $G$ if $\mathcal{L}_{v_iv_j} < 0$ when $v_i$ and $v_i$ are adjacent vertices of $G$ and $\mathcal{L}_{v_iv_j} = 0$ when $v_i$ and $v_j$ are distinct and not adjacent. The 2-degree of the vertex $v_i\in V(G)$, denoted by $S_G(v_i)$, is defined to be the sum of degrees of neighbours of $v_i$, i.e., $S_G(v_i)=\sum_{x\thicksim v_i}d_G(x)$. Let $A(G)$ be the adjacency matrix of a graph $G$. The neighbourhood Laplacian matrix of $G$, which is defined as $\mathcal{L}_{N}(G)={\rm diag}(S_G(v_1),\cdots,S_G(v_n))-A(G)$, is a generalized Laplacian of $G$. Let $\mu _{1}^{\prime}\ge \mu _{2}^{\prime}\ge\ldots \ge\mu _{n}^{\prime}$ be the eigenvalues of $\mathcal{L}_{N}(G)$. In this paper, we first show that $\mathcal{L}_{N}(G)$ is positive semi-definite. \[\Delta_N \leq \mu_1^{\prime}\leq \Delta_N+ \Delta,\] where $\Delta_N=\max\{S_G(x)\mid x \in V(G)\}$ and $\Delta=\max\{d_G(x)\mid x \in V(G)\}$. Finally, we show that if $G$ is a connected graph of order $n$, then $\mu_n^{\prime}$ is a simple eigenvalue of $\mathcal{L}_{N}(G)$ and the corresponding eigenvector can be taken to have all its entries positive.