A study of the generalized outerplanar index of zero-divisor graphs
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A study of the generalized outerplanar index of zero-divisor graphs Zahra Barati1 Received: 2 August 2018 / Accepted: 8 April 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract The purpose of this paper is to explore the question of embedding a zero-divisor graph and its iterated line graphs on the plane such that the resulting graphs are generalized outerplanar graphs. We give a full characterization of all zero-divisor graphs with respect to their generalized outerplanar index. Keywords Zero-divisor graph · Iterated line graph · Planar index · Generalized outerplanar index Mathematics Subject Classification 05C10 · 13M05
1 Introduction Recently, the study of the graphs associated to algebraic structures has been an interesting subject. Various papers were published about properties of these graphs and also the relationships between these graphs and their algebraic structures (see e.g. [3,5–7,10]). This paper is mainly devoted to the study of the zero-divisor graph associated to a commutative ring R. Let R be a commutative ring with identity and Z (R) be the set of all its zero-divisor elements. The zero-divisor graph of R is a simple graph in which the vertices are the nonzero zero-divisors of R, and two distinct vertices x and y are adjacent if and only if x y = 0. This graph is denoted by (R). This definition of (R) was introduced in [5] and many of the most basic features of this graph are investigated by Anderson and Livingston. The original definition appeared in [10] by Beck, where he took all elements of the ring as vertices of the graph. The line graph of a graph G, denoted by L(G), is defined as a graph in which each vertex represents an edge of G and two vertices are adjacent if and only if their corresponding edges share a common endpoint in G. Moreover, the kth iterated line graph of G, denoted by L k (G), is defined as follows. L k (G) = L(L k−1 (G)), L 0 (G) = G and L 1 (G) = L(G).
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Zahra Barati [email protected] Department of Mathematics, Kosar University of Bojnord, Bojnord, Iran
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Z. Barati
Definition 1.1 Let G be a graph. Recall that (a) G is a planar graph if it can be drawn on the plane in such a way that its edges intersect only at their endpoints. (b) G is an outerplanar graph if it can be drawn on the plane without crossings in such a way that all of the vertices belong to the external face of the drawing. (c) G is a generalized outerplanar graph if it can be drawn on the plane in such a way that at least one end-vertex of each edge lies on the external face. By Definition 1.1, we can conclude the following implications for any graph G: outerplanar ⇒ generalized outerplanar ⇒ planar In [13], M. Ghebleh et al. defined the planar index for a graph G as follows: the planar index of a graph G, denoted by ξ(G), is the smallest k such that L k (G) is non-planar and this index equals infinity if L k (G) is planar for all k ≥ 0. In [13], the authors gave a full characterization for graphs with respect to their planar i
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