We study the query complexity of determining if a graph is connected with global queries. The first model we look at is matrix-vector multiplication queries to the adjacency matrix. Here, for an $n$-vertex graph with adjacency matrix $A$, one can query a vector $x \in \{0,1\}^n$ and receive the answer $Ax$. We give a randomized algorithm that can output a spanning forest of a weighted graph with constant probability after $O(\log^4(n))$ matrix-vector multiplication queries to the adjacency matrix. This complements a result of Sun et al.\ (ICALP 2019) that gives a randomized algorithm that can output a spanning forest of a graph after $O(\log^4(n))$ matrix-vector multiplication queries to the signed vertex-edge incidence matrix of the graph. As an application, we show that a quantum algorithm can output a spanning forest of an unweighted graph after $O(\log^5(n))$ cut queries, improving and simplifying a result of Lee, Santha, and Zhang (SODA 2021), which gave the bound $O(\log^8(n))$. In the second part of the paper, we turn to showing lower bounds on the linear query complexity of determining if a graph is connected. If $w$ is the weight vector of a graph (viewed as an $\binom{n}{2}$ dimensional vector), in a linear query one can query any vector $z \in \mathbb{R}^{n \choose 2}$ and receive the answer $\langle z, w\rangle$. We show that a zero-error randomized algorithm must make $\Omega(n)$ linear queries in expectation to solve connectivity. As far as we are aware, this is the first lower bound of any kind on the unrestricted linear query complexity of connectivity. We show this lower bound by looking at the linear query \emph{certificate complexity} of connectivity, and characterize this certificate complexity in a linear algebraic fashion.

翻译：我们研究确定一个图形是否与全球查询相连接的查询复杂性。 我们查看的第一个模型是 矩阵- 矢量递增查询对相邻矩阵。 在这里, 对于一个 $$ 和相邻矩阵矩阵 $A$, 人们可以查询一个矢量 $x $ $ 0. 1 美元 美元, 并接收答案 $Ax 。 我们给出一个随机化算法, 可以在 $( log2) 后, 美元 矩阵- 矢量递增查询 。 这补充了 Sun 和 al. (OCP 2019) 的结果, 该结果提供了随机化的算法, 在 $( log_ 4) 之后, 美元 矩阵- 倍递增查询 。 作为应用程序, 量算法可以输出一个在 $ ( log2) 的直径直径直径直径( = 美元) 直径直径直径直径直径直径直径直图, 直径直径直径直径直径直径直径直径直径直径直调查询。