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Description
Let $G(V,E)$ be a simple graph of order $n$, size $m$ and having the vertex set $V(G)=\{v_1, v_2, \dots, v_n\}$ and edge set $E(G)=\{e_1, e_2,\dots, e_m\}$. The adjacency matrix $A=(a_{ij})$ of $G$ is a $(0, 1)$-square matrix of order $n$ whose $(i,j)$-entry is equal to 1 if $v_i$ is adjacent to $v_j$ and equal to 0, otherwise. Let $D(G)={diag}(d_1, d_2, \dots, d_n)$ be the diagonal matrix associated to $G$, where $d_i=\deg(v_i),$ for all $i=1,2,\dots,n$. The matrix $L(G)=D(G)-A(G)$ is called the Laplacian matrix and its eigenvalues are called the Laplacian eigenvalues of the graph $G$. Let $0=\mu_n\leq\mu_{n-1}\leq \dots \leq\mu_1$ be the Laplacian eigenvalues of $G$ and $S_k(G)=\sum\limits_{i=1}^{k}\mu_i$, $k=1,2,\dots,n$ be the sum of $k$ largest Laplacian eigenvalues of $G$. For any $k$, $k=1,2,\dots,n$, A. Brouwer conjectured that $S_k(G)=\sum\limits_{i=1}^{k}\mu_i\leq m+{k+1 \choose 2}$. We discuss the recent developments on the Brouwer's conjecture.