Characterization by Intersection Graph of Some Families of Finite Nonsimple Groups
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Czechoslovak Mathematical Journal
19 pp
Online first
CHARACTERIZATION BY INTERSECTION GRAPH OF SOME FAMILIES OF FINITE NONSIMPLE GROUPS Hossein Shahsavari, Behrooz Khosravi, Tehran Received June 3, 2019. Published online September 17, 2020.
Abstract. For a finite group G, Γ(G), the intersection graph of G, is a simple graph whose vertices are all nontrivial proper subgroups of G and two distinct vertices H and K are adjacent when H ∩ K 6= 1. In this paper, we classify all finite nonsimple groups whose intersection graphs have a leaf and also we discuss the characterizability of them using their intersection graphs. Keywords: intersection graph; leaf; nonsimple group; characterization MSC 2020 : 05C25, 20D99
1. Introduction Throughout this paper all graphs are finite, undirected, with no loops and no multiple edges. The vertex set and the edge set of a graph Γ are denoted by V (Γ) and E(Γ), respectively. A complete graph is a graph in which all vertices are adjacent and a null graph is a graph with no edges. For a vertex v, deg(v) is the number of vertices adjacent to v and is called the degree of v. A vertex of degree 1 is called a leaf. By nl(Γ), we denote the number of leaves of a graph Γ. A path between two distinct vertices u, v ∈ V (Γ), is defined as a sequence of distinct vertices u = v0 , v1 , . . . , vn = v such that {vi , vi+1 } ∈ E(Γ) for 0 6 i 6 n − 1, and n is called the length of this path. The length of a shortest path between two distinct vertices u and v is called the distance between them and is denoted by d(u, v). In case there is no path connecting u and v, we define d(u, v) to be infinite. A connected graph is a graph in which there exists a path between each two distinct vertices. For a vertex v ∈ V (Γ), the neighbourhood of v is denoted by N (v) and is defined as N (v) = {u ∈ V (Γ) : {u, v} ∈ E(Γ)}. If S is a nonempty subset of V (Γ), then the S neighbourhood of S is defined as N (S) = N (v). We say two vertices u and v v∈S
DOI: 10.21136/CMJ.2020.0250-19
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are coneighbour when N (u) \ {v} = N (v) \ {u}. By this definition, two nonadjacent leaves are coneighbour when they are both adjacent to a vertex v0 . For a finite group G, the intersection graph of G, denoted by Γ(G), is an undirected graph whose vertex set consists of all nontrivial proper subgroups of G and two distinct vertices H and K are adjacent when H ∩ K 6= 1. Csákány and Pollák in [2] introduced the intersection graph of a finite group. Later, some authors determined finite groups with disconnected intersection graphs, see [9], K3,3 -free intersection graphs, see [4], planar intersection graphs, see [6] and triangle-free intersection graphs, see [1]. Also the intersection graphs of abelian groups are discussed in [5] and [11]. The authors in [7] classified all finite groups with regular intersection graphs. Recently the authors in [8] classified all finite simple groups whose intersection graphs have a leaf and as a consequence, it is proved that these groups are uniquely determined by their intersection graphs. We say a
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