There is a great work of Alizadeh that in section 4 speaks about Minimizing sum of the first few(k-largest) eigenvalues of a symmetric matrix. Instead of a symmetric model we use the weighted Laplacian $(L_c(G) = \displaystyle \sum_{ij \in E} c_{ij}E_{ij})$of graph $G = (N, E)$. Then the dual problem is the following model:

\begin{equation*} \begin{array}{llll} \min & k z + {\rm trace~}V \\ \rm{s.t.} & Iz + V - \sum_{ij \in E} c_{ij}E_{ij} \succeq 0 , \\ % U & \sum_{ij \in E} c_{ij} = 1 ,\\ % V & c \geq 0, ~ V \succeq 0 . \end{array} \end{equation*}

Again, we write the dual of the above model to obtain this model:

\begin{equation*} \begin{array}{llll} \max & x\\ \rm{s.t.} & {\rm trace~} Y = k, \\ & x \leq \langle E_{ij}, Y \rangle ~~~~ \textrm{for } ij \in E,\\ & 0 \preceq Y \preceq I. \end{array} \end{equation*}

Because $Y$ is a positive semidefinite matrix, i.e., $Y \succeq 0$, one property of such a matrices allows us to use the Gram representation: $Y = V^T V$ with $V \in \mathbb{R}^{n \times n}$. If $v_i$ denotes the $i$-th column of $V$ then how can one write condition $Y \preceq I$ in terms of $v_i$ which $I$ is identity matrix with all diagonal entries equal to $1$?

For example the gram representation of first constraint is: $${\rm trace~} Y = \sum_{i \in N} \| v_i \|^2 = k$$