# Terminology: What is a "support graph"?

Is a "support graph" the same thing as the unweighted graph corresponding to a matrix, if we treat that matrix as if it were a weighted adjacency matrix (with possibly negative weights)?

In particular the graph would be undirected if and only if the corresponding matrix was symmetric?

The term was used repeatedly by a computer science professor in a lecture they gave, but it seemed to be assumed that the audience would know what this refers to.

All of the Google results for "support graph" (in quotes, so only matches with that exact phrase) which I read through assumed that the reader already knew what the term meant.

Note: The term might be specific to the field of graphical models, since that was the subject of the lecture. Because it might be specific to that field, I decided to post this question here instead of one of the StackExchange sites dedicated solely to computer science -- please let me know if this was a mistake so I can delete this question and post it again on the more appropriate site.

The answer can be found in a recent paper co-authored by the professor giving the lecture, specifically Graphical Lasso and Thresholding: Equivalence and Closed-form Solutions by Salar Fattahi and Somayeh Sojoudi. It seems that my guess was close but not quite accurate.

Given a symmetric matrix $S \in \mathbf{S}^d$ [$\mathbf{S}^d$ is the set of all $d \times d$ symmetric matrices], the support graph or sparsity graph of $S$ is defined as a graph with the vertex set $\mathcal{V}:= \{1,2,\dots, d\}$ and the edge set $\mathcal{E} \subseteq \mathcal{V} \times \mathcal{V}$ such that $(i,j) \in \mathcal{V}$ if and only if $S_{ij} \not= 0$, for every two different vertices $i,j \in \mathcal{V}$. The support graph of $S$ captures the sparsity pattern of the matrix $S$ and is denoted as $supp(S)$.