Abstract

Let G be a simple graph of order n and let μ be an adjacency eigenvalue with multiplicity k≥1. A star complement H for μ in G is an induced subgraph of order n−k with no eigenvalue μ. In this paper, the regular graphs with the complete multipartite graph sK1∪˙Kt‾(s≥2) as a star complement are investigated. More precisely, when μ is a main eigenvalue, we prove that G=Ks+2. When μ is a non-main eigenvalue, we determine the regular graphs with sK1∪˙Kt‾ as a star complement for t=1 or s=3. We show that the complete multipartite graph (s+1)K2‾(s≥2) is the only possible graph under a certain condition, and we raise a conjecture for the general case.

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