I'm reading through this paper, and in Lemma 2.3 on page 4 they use a notation I am unfamiliar with. They say for $(m\times m)$ matrix $\mathbf{A}$ with eigenvector value at the first index, $v_1$ that
$$\Bigg|\sum_{j=1}^m \mathbf{A}_{1,j} v_j\Bigg| = \Bigg|\sum_{j \sim 1} \mathbf{A}_{1,j} v_j\Bigg|$$
What does the $\sim$ symbol mean here?
[Edit:] fixed paper link (again)
[Edit:] More info: $\mathbf{A}$ is not exactly an adjacency matrix in the traditional sense. It is a symmetrical $(m\times m)$ matrix whose entries are all in $\{-1,0,1\}$
I've never seen the notation before, but it looks to me like adjacency of vertices (but at the level of indices, that is, $i\sim j$ if $v_i$ and $v_j$ are adjacent in $H$).
In particular, the last inequality in the displayed math from Lemma 2.3 requires that the number of $j$ with $j\sim 1$ is at most the maximum degree $\Delta(H)$, and the equality in the OP holds if $j\not\sim 1$ implies $A_{1,j}=0$ (or $v_j=0$, but we have no control over this).