We can generalize the representation of convolutional filters beyond the chain graph by using arbitrary adjacency matrices $A$ or Laplacians. We can define a generalized polynomial filter as:

$Q_h = \alpha_0I + \alpha_1A + \alpha_2A^2 + \dots + \alpha_KA^K$

When we multiply this filter by a matrix of node features $\mathbf{X} \in \mathbb{R}^{|V| \times m}$, we get:

$Q_h\mathbf{X} = \alpha_0I\mathbf{X} + \alpha_1A\mathbf{X} + \alpha_2A^2\mathbf{X} + \dots + \alpha_KA^K\mathbf{X}$

where $(Q_h\mathbf{X})[u]$ at a given node $u$ corresponds to a feature vector $\in \mathbb{R}^m$ that aggregates information from the node's $K$-hop neighborhood.