I had asked this question in [math.se][1] without any success

Let $A$ be the symmetric $n\times n$ adjacency matrix for a graph where $A_{ij}$ is the positive edge value between node $i$ and $j$ (thus fully connected graph). Among the $n-$ nodes, let $c$ be a given node of interest (called a central node in my problem). Define the matrix \begin{align} B_{ij}=\begin{cases} 0 &,~~i=j \\ \frac{A_{ij}}{A_{ic}A_{jc}} &,~~\text{otherwise} \end{cases} \end{align} Thus the individual elements are proportional to distance between $i$ and $j$ and inversely proportional to distance from the central node. Thus, this term will be high for pairs which are close to central node, but far from each other. I am interested in solving the following optimization problem \begin{align} \max_{x_{ij}}\sum_{i,j}x_ix_j&B_{ij} \\s.t.~~\sum_{i=1}^{n}x_i\leq K ~~,&~~x_i\in\{0,1\} \end{align} Intuitively, I need to select $K$ nodes such that they are as close as possible to the central node yet far apart among themselves?

Is this problem studied in literature? Is it NP-Hard? what are the known practical approaches? I am familiar with the Semidefinite formulation of this. I am interested in knowing if there are graph based approaches. [1]: https://math.stackexchange.com/questions/3375396/is-this-graph-problem-np-hard?noredirect=1#comment6946013_3375396

I had asked this question in math.se without any success

Let $A$ be the symmetric $n\times n$ adjacency matrix for a graph where $A_{ij}$ is the positive edge value between node $i$ and $j$ (thus fully connected graph). Among the $n-$ nodes, let $c$ be a given node of interest (called a central node in my problem). Define the matrix \begin{align} B_{ij}=\begin{cases} 0 &,~~i=j \\ \frac{A_{ij}}{A_{ic}A_{jc}} &,~~\text{otherwise} \end{cases} \end{align} Thus the individual elements are proportional to distance between $i$ and $j$ and inversely proportional to distance from the central node. Thus, this term will be high for pairs which are close to central node, but far from each other. I am interested in solving the following optimization problem \begin{align} \max_{x_{ij}}\sum_{i,j}x_ix_j&B_{ij} \\s.t.~~\sum_{i=1}^{n}x_i\leq K ~~,&~~x_i\in\{0,1\} \end{align} Intuitively, I need to select $K$ nodes such that they are as close as possible to the central node yet far apart among themselves?

Is this problem studied in literature? Is it NP-Hard? what are the known practical approaches? I am familiar with the Semidefinite formulation of this. I am interested in knowing if there are graph based approaches.