Department of Mathematics

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Author: search.filters.author.Kumar, JitenderSubject: search.filters.subject.Enhanced power graph

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    On the difference graph of power graphs of finite groups
    (Taylor & Francis, 2023-11) Kumar, Jitender
    The power graph of a finite group G is the simple undirected graph with vertex set G whose two vertices are adjacent if one is a power of the other. The enhanced power graph of a finite group G is the simple undirected graph whose vertex set is the group G whose two vertices a and b are adjacent if there exists c ∈ G such that both a and b are powers of c. In this paper, we investigate the difference graph Ɗ(G) of a finite group G, which is the difference of the enhanced power graph and the power graph of G with all isolated vertices removed. We first characterize an arbitrary finite group G such that Ɗ(G) is a chordal graph, star graph, dominatable, threshold graph, and split graph. From this, we conclude that the latter four graph classes are equal for Ɗ(G). By applying these results, we classify the nilpotent groups G such that Ɗ(G) belong to the aforementioned five graph classes. This shows that all these graph classes are equal for Ɗ(G) when G is nilpotent. Then, we characterize the nilpotent groups whose difference graphs are cograph, bipartite, Eulerian, planar, and outerplanar. Finally, we consider the difference graph of non-nilpotent groups and determine the values of n such that the difference graphs of the symmetric group Sn and alternating group An are cograph, chordal, split, and threshold.
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    On enhanced power graphs of certain groups
    (World Scientific, 2021) Kumar, Jitender
    The enhanced power graph Pe(G) of a group G is a simple undirected graph with vertex set G and two distinct vertices x,y are adjacent if both x and y belongs to same cyclic subgroup of G. In this paper, we obtain various graph invariants viz. independence number, minimum degree and matching number of Pe(G), where G is the dicyclic group or a class of groups of order 8n. If G is any of these groups, we prove that Pe(G) is perfect and then obtain its strong metric dimension.
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    On the enhanced power graph of a finite group
    (Taylor & Francis, 2020-11) Kumar, Jitender
    The enhanced power graph Pe(G) of a group G is a graph with vertex set G and two vertices are adjacent if they belong to the same cyclic subgroup. In this paper, we consider the minimum degree, independence number, and matching number of enhanced power graphs of finite groups. We first study these graph invariants for Pe(G) when G is any finite group and then determine them when G is a finite abelian p-group, U6n=⟨a,b:a2n=b3=e,ba=ab−1⟩, the dihedral group D2n, or the semidihedral group SD8n. If G is any of these groups, we prove that Pe(G) is perfect and then obtain its strong metric dimension. Additionally, we give an expression for the independence number of Pe(G) for any finite abelian group G. These results along with certain known equalities yield the edge connectivity, vertex covering number, and edge covering number of enhanced power graphs of the respective groups as well.