“School of Mathematics”
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| Paper IPM / M / 18003 | ||||||||||||||||||
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A subset C of the vertex set of a graph Γ is said to be (α,β)-regular if C induces an α-regular subgraph and every vertex outside C is adjacent to exactly β vertices in C. In particular, if C is an (α,β)-regular set in some Cayley sum graph of a finite group G with connection set S, then C is called an (α,β)-regular set of G. By Sq(G) and NSq(G) we mean the set of all square elements and non-square elements of G. As one of the main results in this note, we show that a subgroup H of a finite abelian group G is an (α,β)-regular set of G, for each 0≤α≤|NSq(G)∩H| and 0≤β≤L(H), where L(H)=|H|, if Sq(G)⊆H and L(H)=|NSq(G)∩H|, otherwise. As a consequence we easily get that H is a (0,1)-regular, if and only if either Sq(G)⊆H or NSq(G)∩H≠∅. The proof of this result is given by X. Ma, K. Wang, and Y. Yang in 2022, in a longer method. Also, X. Ma, M. Feng, and K. Wang in 2020, gave a sufficient and necessary condition for a subgroup H to be a (0,1)-regular subgroup of an abelian group G. Our new result makes that result much easier to gain.
Also, we consider the dihedral group G=D2n and for each subgroup H of G, by giving an appropriate connection set S, we determine each possibility for (α,β), where H is an (α,β)-regular set of G.
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