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<p>An algebraic structure, once associated with a suitable graph, can be studied through the tools of graph theory. In recent years, algebraic graph theorists have been interested with this issue (especially when the algebraic structure is a group). Bianchi et al. introduced the permutability graph of nonnormal subgroups in 1995. This graphwas then modified to the permutability graph of subgroups, with a vertex set to all proper subgroups of <i>G</i> and the same condition for joining two vertices <i>H</i> and <i>K</i>, <i>HK</i> = <i>KH</i>. Further generalization was done by Muhie et al. by examining the nonpermutability graph of subgroups. Recent research on the spectral properties of the nonpermutability graph of subgroups has focused on <i>F</i><sub>2</sub>(<i>G</i>) and sd(<i>G</i>), revealing some new combinatorial formulas involving adjacency and Laplacian matrices. On the other hand, the subgroup commutativity degree sd(<i>G</i>) of <i>G</i> is the probability of finding two commuting subgroups in <i>G</i> at random, and the factorization number <i>F</i><sub>2</sub>(<i>G</i>) of a finite group <i>G</i> is the number of all possible factorizations of <i>G</i> = <i>HK</i> as a product of its subgroups <i>H</i> and <i>K</i>. In this work, we present some spectral features of permutability graphs of subgroups, including <i>F</i><sub>2</sub>(<i>G</i>) and sd(<i>G</i>). With our new method, we may move forward without assuming that sd(<i>G</i>) ≠ 1.</p>

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Spectral properties of the permutability graph of subgroups

  • 期刊论文

  • 发表日期：

    2026年1月1日

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  • Anteneh Alemu Ali,
  • Seid Kassaw Muhie,
  • Daniele Ettore Otera

摘要

An algebraic structure, once associated with a suitable graph, can be studied through the tools of graph theory. In recent years, algebraic graph theorists have been interested with this issue (especially when the algebraic structure is a group). Bianchi et al. introduced the permutability graph of nonnormal subgroups in 1995. This graphwas then modified to the permutability graph of subgroups, with a vertex set to all proper subgroups of G and the same condition for joining two vertices H and K, HK = KH. Further generalization was done by Muhie et al. by examining the nonpermutability graph of subgroups. Recent research on the spectral properties of the nonpermutability graph of subgroups has focused on F2(G) and sd(G), revealing some new combinatorial formulas involving adjacency and Laplacian matrices. On the other hand, the subgroup commutativity degree sd(G) of G is the probability of finding two commuting subgroups in G at random, and the factorization number F2(G) of a finite group G is the number of all possible factorizations of G = HK as a product of its subgroups H and K. In this work, we present some spectral features of permutability graphs of subgroups, including F2(G) and sd(G). With our new method, we may move forward without assuming that sd(G) ≠ 1.