图之字形积的连通性和同构性
Connectedness and isomorphism properties of the zig-zag product of graphs
摘要：
本文研究了两个图之字形积的连通性和同构问题。给出了两个图之字形积连通的一个充分条件，从而把对一个新图的连通性的研究归结为仅依赖于图积的第二因子的问题.我们表明，当第二个因素是一个循环图，研究的同构问题的锯齿形产品是等价于研究同样的问题相关联的伪置换图。后者是以一种自然的方式定义的，由一个结构推广了经典的替代产品，其程度是小于的程度之字形产品图。研究了两类特殊的积：完全图与圈图的Z字形积，以及$4$ -正则图与长为$4$的圈图的Z字形积.此外，还考虑了与自相似群作用有关的Schreier图理论中的一个例子：完全确定了图的乘积，并给出了它们的谱分析.
英文摘要：
In this paper we investigate the connectedness and the isomorphism problems for zig-zag products of two graphs. A sufficient condition for the zig-zag product of two graphs to be connected is provided, reducing to the study of the connectedness property of a new graph which depends only on the second factor of the graph product. We show that, when the second factor is a cycle graph, the study of the isomorphism problem for the zig-zag product is equivalent to the study of the same problem for the associated pseudo-replacement graph. The latter is defined in a natural way, by a construction generalizing the classical replacement product, and its degree is smaller than the degree of the zig-zag product graph. Two particular classes of products are studied in detail: the zig-zag product of a complete graph with a cycle graph, and the zig-zag product of a $4$-regular graph with the cycle graph of length $4$. Furthermore, an example coming from the theory of Schreier graphs associated with the action of self-similar groups is also considered: the graph products are completely determined and their spectral analysis is developed.
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