Abstract Given a bipartite graph G = ( V , E ) with bipartition V = A ∪ B , where | A | = | B | . We present a new construction of pairs of cospectral bipartite graphs by “rotating” G in two different ways. More precisely, let l and k be two positive integers with l ≠ k . Take l copies of A (resp. B ) and k copies of B (resp. A ), say A 1 , A 2 , … , A l and B 1 , B 2 , … , B k (resp. B 1 ′ , B 2 ′ , … , B l ′ and A 1 ′ , A 2 ′ , … , A k ′ ). Let Γ (resp. Γ ′ ) be the graph with vertex set V ( Γ ) = A 1 ∪ A 2 ∪ ⋯ ∪ A l ∪ B 1 ∪ B 2 ∪ ⋯ ∪ B k (resp. V ( Γ ′ ) = B 1 ′ ∪ B 2 ′ ∪ ⋯ ∪ B l ′ ∪ A 1 ′ ∪ A 2 ′ ∪ ⋯ ∪ A k ′ ), such that the vertices of Γ (resp. Γ ′ ) are adjacent if and only if they are adjacent in G . We show that Γ and Γ ′ are cospectral with respect to the adjacency matrix, as well as the normalized Laplacian matrix. Moreover, we give a simple necessary and sufficient condition for Γ and Γ ′ to be non-isomorphic. The main ingredient of the proof involves a use of Hall’s Marriage Theorem.

中文翻译：

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构建共谱二分图

摘要 给定一个二分图 G = ( V , E ) 且二分图 V = A ∪ B ，其中 | 一个 | = | 乙 | . 我们通过以两种不同的方式“旋转” G 来呈现一对共谱二部图的新构造。更准确地说，让 l 和 k 是两个正整数，其中 l ≠ k 。取 A (resp. B ) 的 l 个副本和 B (resp. A ) 的 k 个副本，比如 A 1 , A 2 , ... , A l 和 B 1 , B 2 , ... , B k (resp. B 1 ′ , B 2 ′ , … , B l ′ 和 A 1 ′ , A 2 ′ , … , A k ′ )。设 Γ (resp. Γ ′ ) 是顶点集 V ( Γ ) = A 1 ∪ A 2 ∪ ⋯ ∪ A l ∪ B 1 ∪ B 2 ∪ ⋯ ∪ B k (相应的 V ( Γ ′ ) = B 1 ′ ∪ B 2 ′ ∪ ⋯ ∪ B l ′ ∪ A 1 ′ ∪ A 2 ′ ∪ ⋯ ∪ A k ′ )，使得 Γ 的顶点（分别为 Γ ′ ）相邻当且仅当它们在G 。我们证明 Γ 和 Γ ' 是关于邻接矩阵以及归一化拉普拉斯矩阵的共谱。而且，我们给出一个简单的充分必要条件 Γ 和 Γ ′ 是非同构的。证明的主要成分涉及霍尔婚姻定理的使用。