We prove an analogue of Alon’s spectral gap conjecture for random bipartite, biregular graphs. We use the Ihara–Bass formula to connect the non-backtracking spectrum to that of the adjacency matrix, employing the moment method to show there exists a spectral gap for the non-backtracking matrix. A by-product of our main theorem is that random rectangular zero-one matrices with fixed row and column sums are full rank with high probability. Finally, we illustrate applications to community detection, coding theory, and deterministic matrix completion.

中文翻译：

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随机二部双正则图和应用中的光谱间隙

我们证明了随机二分双正则图的 Alon 谱隙猜想的类似物。我们使用 Ihara-Bass 公式将非回溯谱与邻接矩阵的谱联系起来，采用矩方法表明非回溯矩阵存在谱间隙。我们的主要定理的一个副产品是具有固定行和列和的随机矩形零一矩阵很有可能是满秩的。最后，我们说明了社区检测、编码理论和确定性矩阵完成的应用。