We provide a process to extend any bipartite diametrical graph of diameter 4 to an S ‐graph of the same diameter and partite sets. For a bipartite diametrical graph of diameter 4 and partite sets U and W , where 2 m = | U | ≤ | W |, we prove that 2 m is a sharp upper bound of | W | and construct an S ‐graph G (2 m , 2 m ) in which this upper bound is attained, this graph can be viewed as a generalization of the Rhombic Dodecahedron. Then we show that for any m ≥ 2, the graph G (2 m , 2 m ) is the unique (up to isomorphism) bipartite diametrical graph of diameter 4 and partite sets of cardinalities 2 m and 2 m , and hence in particular, for m = 3, the graph G (6, 8) which is just the Rhombic Dodecahedron is the unique (up to isomorphism) bipartite diametrical graph of such a diameter and cardinalities of partite sets. Thus we complete a characterization of S ‐graphs of diameter 4 and cardinality of the smaller partite set not exceeding 6. We prove that the neighborhoods of vertices of the larger partite set of G (2 m , 2 m ) form a matroid whose basis graph is the hypercube Q m . We prove that any S ‐graph of diameter 4 is bipartite self complementary, thus in particular G (2 m , 2 m ). Finally, we study some additional properties of G (2 m , 2 m ) concerning the order of its automorphism group, girth, domination number, and when being Eulerian.
The complementary join of a graph G is introduced in this paper as the join G+G of G and its complement considering them as vertex-disjoint graphs. The aim of this paper is to study some properties and some graph invariants of the complementary join of a graph. We find the diameter, the radius and the domination number of G + G and determine when G + G is self-centered. We obtain a characterization of the Eulerian complementary joins, and show that the complementary join of a nontrivial graph is Hamiltonian. We give the clique and independence numbers of G + G in terms of the clique and independence numbers of G. We conclude this paper by determining the chromatic number, the L(2, 1)-labeling number, the locating chromatic number and the partition dimension of the complementary join of a star.
In this paper, we study some operations which produce new divisor graphs from old ones. We prove that the contraction of a divisor graph along a bridge is a divisor graph. For two transmitters (receivers) u and v in some divisor orientation of a divisor graph G, it is shown that the merger G |u,v is also a divisor graph. Two special types of vertex splitting are introduced, one of which produces a divisor graph when applied on a cutvertex of a given divisor graph, while the other is applied on a transmitter (receiver) in some divisor orientation of a given divisor graph and produces a divisor graph.
This article introduces the concept of a D-levels locally homogeneous graph which forms a special type of self-centered graphs. Some results relating D-levels locally homogeneous graphs to diametrical graphs (which are also self-centered) are obtained. Diametrical graphs of small order (≤ 8) are shown to be D-levels locally homogeneous. When the diameter is 2, it is shown that the concept of a D-levels locally homogeneous graph and that of a diametrical graph coincide. Bipartite diametrical graphs of diameter 3 and regular S-graphs of diameter 4 are characterized as D-levels locally homogeneous graphs. We also prove that the vertex-transitive graphs are precisely the totally D-levels locally homogeneous graphs.
Abstract This article studies the zero divisor graph for the ring of Gaussian integers modulo n, Γ (ℤ n [i]). For each positive integer n, the number of vertices, the diameter, the girth and the case when the dominating number is 1 or 2 is found. Complete characterizations, in terms of n, are given of the cases in which Γ (ℤ n [i]) is complete, complete bipartite, planar, regular or Eulerian. Key Words: Bipartite graphComplete graphDiameterEulerian graphGaussian integersGirthGraphPlanar graphZero divisor graph2000 Mathematics Subject Classification: 05C7513A99 Notes Communicated by I. Swanson.
Let R be a commutative finite principal ideal ring with unity, and let G(R) be the simple graph consisting of nontrivial proper ideals of R as vertices such that two vertices I and J are adjacent if they have nonzero intersection. In this paper we continue the work done by Abu Osba. We calculate the radius, eccentricity, domination number, independence number, geodetic number, and the hull number for this graph. We also determine when G(R) is chordal. Finally, we study some properties of the complement graph of G(R) .
Abstract This paper is a continuation for the study of the zero-divisor graph for the ring of Gaussian integers modulo n, Γ(ℤ n [ i ]) in [ 8 ] (Emad Abu Osba, Salah Al-Addasi and Nafez Abu Jaradeh. Zero divisor graph for the ring of Gaussin integers modulo n. Comm. Algebra 36(10) (2008), 3865–3877). It is investigated, when is Γ(ℤ n [ i ]) locally H, Hamiltonian or bipartite graph? A full characterisation for the chromatic number and the radius is also given.
A diametrical graph G is said to be symmetric if for all u , v ∈ V ( G ), where is the buddy of u . If moreover, G is bipartite, then it is called an S ‐graph. It would be shown that the Cartesian product K 2 × C 6 is not only the unique S ‐graph of order 12 and diameter 4, but also the unique symmetric diametrical graph of order 12 and diameter 4.
