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46th Annual Iranian Mathematics Conference 25-28 August 2015 Yazd University

Talk d-self center graphs and graph operations pp.: 1–4

d-self Center Graphs and Graph Operations

Yasser Alizadeh Ehsan Estaji ∗ Hakim Sabzevari University Hakim Sabzevari University

Abstract Let G be a simple connected graph. The graph G is called d-self center if it’s vertices are of eccentricity d. In this paper, some self center composite graphs are investigated. Some mathematical properties of self center graphs is investigated. It is proved that a self center graph is 2-connected. Some infinite family of asymmetric self center graphs is constructed.

Keywords: eccentricity, d-self center graph, composite graphs Mathematics Subject Classification [2010]: 05C12, 05C76, 05C90

1 Introduction

All considered graphs are simple and connected. between two vertices is defined as usual length of shortest path connecting them. Eccentricity of v is denoted by ε(v) is the maximum distance between v and other vertices. The maximum and minimum eccentricity among all vertices of G are called diameter of G, diam(G) and radius of G, rad(G) respectively. The Center of G, C(G) is the set of vertices of rad(G). Let G be a simple connected graph. The graph G is called d-self center if it’s vertices are of eccentricity d. Center of graph G is the set of vertices of minimum eccentricity. Then the Center of a self center graph contains all vertices of the graph. In a series of paper, topological indices based on eccentricity of vertices were studied and for some family of molecular graphs such indices were calculated. For more information, we refer the reader to [1, 2, 3, 4, 5, 6, 8, 7, 10]. In this paper, self center graphs under some graph operations is studied. It is proved that a self center graph is 2-connected. By graph operations, some asymmetric self center graphs is constructed.

2 Main Results

n As example complete graphs Kn, cycles Cn and sierpinski graphs Sk are three well-known family of self center graphs. A graph G is called vertex- transitive if for given vertices u and v there is an auto-morphism of G, f such that f (u) = v. For example the complete graphs and graphs and Petersen graph are vertex transitive graphs. Since distance between vertices and eccentricity are invariant under auto morphism of graphs, then the

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46th Annual Iranian Mathematics Conference 25-28 August 2015 Yazd University

Talk d-self center graphs and graph operations pp.: 2–4

n vertex transitive graphs are self center but the reverse is not true. The Sierpinski graphs Sk are a family of self center graphs but non vertex transitive. A that is edge- transitive but not vertex-transitive is called a semi-. The (with 54 vertices), the Tutte 12-cage (with 126 vertices) are two namely semi-symmetric and self center graph and the Folkman graph (with 20 vertices) and the (with 112 vertices) are other semi-symmetric graph but non self center graph. It seems an interesting problem to characterize the self center semi-symmetric graphs. A self center graph with n 3 vertices is a block graph or 2-connected graph. ≥ Theorem 2.1. Let G be a self center graph with n 3 vertices. Then G is 2-connected. ≥ There are some self center graphs such as cycles that are not 3-connected graph. Let Aut(G) be the group automorphism of graph G. Orbit of vertex v is denoted by Orb(v) and defined as Orb(v) = f (v) f Aut(G) . The vertices of Orb (v) have the same eccentricity. | ∈ A graph G is vertex transitive if and only if G has exactly one orbit. The following example illustrated in Figure 1. is a graph with 7 orbits but all vertices have a same eccentricity then the graph is self center.

Figure 1: asymmetric graph with 7 orbits

Proposition 2.2. [9] If G is a 2-self center graph on n 5 vertices then G has at least 2n 5 ≥ − edges.

3 Composite graphs

In this section some self centered graph arised from graph operation are presented. We start by join of graphs.

Theorem 3.1. Let G1 and G2 be two simple connected graphs. Then G1 + G2 is self center graph if and only if G1 and G2 are self center graphs.

Proposition 3.2. For any n 4 there is a family of 2-self center graphs on n vertices. ≥ It is enough to consider K + K where m + p = n and m, p 2. We have a similar m p ≥ statement about cartesian product of graphs.

Theorem 3.3. Let G and H be two simple connected graphs. Then G G2 is self center graph if 1 × and only if G1 and G2 are self center graphs.

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46th Annual Iranian Mathematics Conference 25-28 August 2015 Yazd University

Talk d-self center graphs and graph operations pp.: 3–4

n Corollary 3.4. Gi is self center iff each Gi is self center for 1 i n. i=1 ≤ ≤ Q Corollary 3.5. Ther is an infinite family of non vertex transitive self centered .

Consider the powers of a non-vertex transitive self center graph such as the Gray graph or the Tutte 12-cage graph. For any two simple connected graph with at least two vertices, we can construct a self center graph by symmetric difference operation of the graphs.

Theorem 3.6. Let G and H be two simple connected graph with at least two vertices. Then the symmetric difference G H is 2-self center. ⊕ It is easy to see that the disjunction of two complete graphs is a complete graph and vice versa. In the case radius of both G and H is at least 2 the disjunction G H is 2-self ∨ center.

Theorem 3.7. Let G and H be two simple connected graph with radius at least 2. Then the disjunction G H is 2-self center. ∨

References

[1] Y. Alizadeh, M. Azari, T. Dosliˇ c,´ Computing the eccentricity-related invariants of single-defeat carbon nanocones, J. Comput. Theoret. Nanosci., to appear.

[2] M. Azari, A. Iranmanesh, Computing the eccentric-distance sum for graph opera- tions, Disc. Appl. Math. 161 (2013) 2827-2840.

[3] N. De, On Eccentric Connectivity Index and Polynomial of Thorn Graph, Applied Mathematics, 3 (2012) 931-934.

[4] N. De, Sk. Md. A. Nayeem, A. Pal, Total Eccentricity Index of the Generalized Hierarchical Product of Graphs, Int. J. Appl. Comput. Math 2014.

[5] T. Dosliˇ c,´ A. Graovac, O. Ori, Eccentric Connectivity Index of Hexagonal Belts and Chains, MATCH Commun. Math. Comput. Chem. 65 (2011) 745-752.

[6] T. Dosliˇ c,´ M. Saheli, Augmented Eccentric Connectivity Index, Miskolc Mathematical Notes. 12 (2011) 149-157.

[7] M. Ghorbani, M.A Hosseinzadeh, new version of Zagreb indices, Filomat. 26:1 (2012) 93-100.

[8] A. Iranmenesh, Y. Alizadeh. Eccentric Connectivity Index of HAC5C7[p,q] and HAC5C6C7[p,q] Nanotubes, MATCH Commun. Math. Comput. Chem. 69 (2013) 175-182.

[9] Z. Stanic, Some notes on minimal self centered graphs, AKCE J. Graphs. Combin. 7 (2010) 97-102

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46th Annual Iranian Mathematics Conference 25-28 August 2015 Yazd University

Talk d-self center graphs and graph operations pp.: 4–4

[10] Z. Yarahmadi, S. Moradi, The Center and Periphery of Composite Graphs, IJMC. 5 (2014) 35-44.

Email: [email protected] Email: [email protected]

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