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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2008, Article IDΒ 468583, 12 pages
http://dx.doi.org/10.1155/2008/468583
Research Article

Bipartite Diametrical Graphs of Diameter 4 and Extreme Orders

1Mathematics Department, Faculty of Science, Hashemite University, Zarqa 150459, Jordan
2Mathematics Department, Faculty of Science, University of Jordan, Amman 11942, Jordan

Received 9 July 2007; Accepted 5 December 2007

Academic Editor: PeterΒ Johnson

Copyright Β© 2008 Salah Al-Addasi and Hasan Al-Ezeh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We provide a process to extend any bipartite diametrical graph of diameter 4 to an -graph of the same diameter and partite sets. For a bipartite diametrical graph of diameter 4 and partite sets and , where , we prove that is a sharp upper bound of and construct an -graph 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 , the graph is the unique (up to isomorphism) bipartite diametrical graph of diameter 4 and partite sets of cardinalities and , and hence in particular, for , the graph 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 -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 form a matroid whose basis graph is the hypercube . We prove that any -graph of diameter 4 is bipartite self complementary, thus in particular . Finally, we study some additional properties of concerning the order of its automorphism group, girth, domination number, and when being Eulerian.

1. Introduction

There are frequent occasions in which graphs with a lot of symmetries are required. One such family of graphs is that of -graphs. -graphs form a special type of symmetric diametrical graphs which are one of the most interesting classes of diametrical graphs, see [1, 2], as well as of -pairable graphs introduced in [3]. Diametrical graphs were studied under different names, see [2, 4–6].

All graphs we consider are assumed to be finite, connected, and to have no loops or multiple edges. For undefined notions and terminology, we use [7].

Two vertices, , and of a nontrivial connected graph are said to be diametrical if A nontrivial connected graph is called diametrical if each vertex of has a unique diametrical vertex , the vertex is called the buddy of , see [4] and [5]. A diametrical graph is called minimal if removing any edge of produces a graph which is not diametrical of the same diameter. A diametrical graph in which whenever is called harmonic. A diametrical graph is called maximal if there is no diametrical graph with the same vertex set and diameter as such that contains properly. A diametrical graph is called symmetric if for all , see [2], that is, a diametrical graph is symmetric if and only if for any , where denotes the interval of consisting of all vertices on shortest -paths, see [5]. A symmetric diametrical graph is called an antipodal graph, and a bipartite antipodal graph is called an -graph, see [1].

It is shown in [8] that the concepts of symmetric and maximal are equivalent in the class of bipartite diametrical graphs of diameter . We provide a process to extend any given bipartite diametrical graph of diameter to a maximal diametrical graph and hence an -graph of the same diameter and partite sets.

For a bipartite diametrical graph of diameter and partite sets and with , it is given in [8] that . In this paper, we improve this upper bound by proving that , where . To show that is a sharp upper bound, we construct a bipartite diametrical graph, which we denote by , of diameter in which this maximum value of is attained. In particular, when , the graph is indeed just the Rhombic Dodecahedron which is sometimes also called the Rhomboidal Dodecahedron, see [9]. The Rhombic Dodecahedron appears in some applications in natural sciences, see, for example, [10, 11]. It is the convex polyhedron depicted in Figure 1 and can be built up by placing six cubes on the faces of a seventh, taking the centers of outer cubes and the vertices of the inner as its vertices, and joining the center of each outer cube to the closer four vertices of the inner. The number is a sharp lower bound of which is previously known through a graph constructed in [8], which we denote in this paper by For , it is proven in [12] that, up to isomorphism, the graph is the unique symmetric diametrical graph of order 12 and diameter 4. For larger , needs not be the unique symmetric diametrical bipartite graph of diameter 4. We give an example of an -graph with the same cardinalities of partite sets as those of but not isomorphic to On the other hand, we prove that, for any , the graph is not only the unique (up to isomorphism) -graph of diameter 4 and partite sets of cardinalities and , but also is the unique bipartite diametrical graph of such a diameter and partite sets. Consequently, the Rhombic Dodecahedron is the unique diametrical graph of diameter and maximum order where the smaller partite set has cardinality . Joining this to the main result of [12], our result about the upper bound of the cardinality of the larger partite set completes a characterization of -graphs of diameter and cardinality of the smaller partite set not exceeding

68583.fig.002
Figure 1: The Rhombic Dodecahedron.

Matroids constitute an important unifying structure which has many equivalent definitions. According to one of these definitions, a matroid on a finite set is a collection of subsets of called bases, which satisfies the exchange property: for all and there exists such that It is well known that all the bases of a matroid have the same cardinality. The basis graph of a matroid is the graph whose vertices are the bases of the matroid, where two vertices are adjacent if they differ by a single exchange, that is, if the symmetric difference has exactly two elements. The basis graphs faithfully represent their matroids, see [13] and references therein; thus studying the basis graph amounts to studying the matroid itself. We prove that the neighborhoods of the vertices of the larger partite set of form a matroid whose basis graph is the hypercube . The hypercubes together with the Rhombic Dodecahedra appear as the cells in the tight span of a totally split-decomposable metric which is used in the field of phylogenetic analysis, see [14].

A bipartite graph with partite sets and is called bipartite self complementary if is a subgraph of which is isomorphic to its complement with respect to the complete bipartite graph with the same partite sets, see [15]. We show that the graphs and are bipartite self complementary and study some of their properties concerning girth, domination number, order of automorphism group, and when being Eulerian.

2. A Sharp Upper Bound of the Cardinality of the Larger Partite Set

One basic property of any bipartite diametrical graph of even diameter is that each of the partite sets has an even number of elements, as stated in Lemma 2.1, see [8].

Lemma 2.1. If is a bipartite diametrical graph of even diameter and partite sets and then for each both and belong to the same partite set, and hence and are even.

In particular, when , each partite set of consists of a vertex , its buddy and those vertices equidistant to and .

It is shown in [2] that symmetric diametrical graphs form a proper special class of maximal diametrical graphs. But the two concepts coincide in the class of bipartite diametrical graphs of diameter according to the following fact which was proven in [8].

Lemma 2.2. Let be a bipartite diametrical graph of diameter 4. Then the following are equivalent:
(i) is maximal in the class of bipartite diametrical graphs with the same partite sets as (ii) is maximal;(iii) is symmetric.

The following theorem includes a process to extend any given bipartite diametrical graph of diameter to a bipartite diametrical graph of the same diameter and partite sets, which is maximal in the class of bipartite diametrical graphs with the same partite sets, and hence, by the previous lemma, is an -graph.

Theorem 2.3. Let be a bipartite diametrical graph of diameter and partite sets and Then, the following hold:
(i)if and such that the set of vertices is independent, then the graph is bipartite diametrical of the same diameter and partite sets as (ii)if but then the graph is bipartite diametrical of the same diameter and partite sets as (iii)if is a graph obtained from by repeated applications of (i) and (ii) until none of them is applicable, then is an -graph of the same diameter and partite sets as

Proof. (i) Assume to the contrary that is either not diametrical or has diameter less than Then there exists a vertex such that . But, since is also bipartite with the same partite sets as and both belong to the same part of , then is even. Hence . Thus there exists a -path in containing the new edge and having length 2. Then either and is or and is , which gives a contradiction in both cases since . Therefore, is a diametrical graph of diameter 4.
(ii) Let such that with and where and are the partite sets of . Then because otherwise , which is not the case. Similarly, . Now, we will show by contradiction that is diametrical of diameter 4. Assume not, then there exists a vertex such that . Clearly, is bipartite with the same partite sets as since and . But both and belong to the same part of , hence is even, and so . Thus, there exists a -path in containing the new edge and having length 2. Consequently, either and is or and is , which gives a contradiction in both cases since . Therefore, is bipartite diametrical of diameter 4.
(iii) Let be a graph obtained from by addition of edges according to (i) or (ii) until none of them can be applied. By Lemma 2.2, it would be enough to show that is maximal in the class of bipartite diametrical graphs of diameter and partite sets and Assume to the contrary that there exist two nonadjacent vertices and of such that still diametrical of diameter Then, since (ii) is no more applicable on we have But also (i) is not applicable on so the set is not independent, therefore either or Consequently, either or a contradiction.

The only bipartite diametrical graph of diameter in which the smaller partite set has cardinality less than is the -cycle as Lemma 2.4 (proven in [8]) says.

Lemma 2.4. If is a bipartite diametrical graph of diameter 4 and partite sets and with , then unless .

For a bipartite diametrical graph of diameter 4 and partite sets and with we have , and whenever , we must have . It was shown in [8] that if , then where . Thus for , we have . This is not a sharp upper bound, as shown at the end of this section. In order to obtain a sharp upper bound, we start by the following lemma.

Lemma 2.5. Let and be distinct vertices of a diametrical graph of diameter . Then

Proof. If , then since . Suppose that and assume to the contrary that . Any shortest path joining and must include a vertex from ; let be such a vertex in such a path. Then But and is the only vertex of at distance from , hence . Therefore, contradicting the uniqueness of the buddy vertex of .

In particular, two distinct vertices in a diametrical graph of diameter at least 3 cannot have the same neighborhood.

Before tending to give an upper bound of , let us recall the following fact proven in [8].

Lemma 2.6. Every vertex of an -graph of diameter 4 is adjacent to exactly half of the vertices in the partite set of not containing .

We are now in a position to prove our improvement of the upper bound of .

Theorem 2.7. Let be a bipartite diametrical graph of diameter 4 and partite sets and , where . Then, .

Proof. By Theorem 2.3, there exists an -graph with the same diameter and partite sets as which has as a subgraph. Then, by Lemma 2.6, each vertex has exactly neighbors in . Clearly, cannot contain a vertex and its buddy. But Lemma 2.5 implies that the sets , are mutually distinct. Hence, must be less than or equal to the number of ways of choosing single representatives from each of two-element sets; clearly that number is .

Now, if is a bipartite diametrical graph of diameter and partite sets and where , then by Lemma 2.1, for some positive integer . Since , we must have . In the next section, we will construct for each integer a bipartite diametrical graph of diameter and partite sets and with and , which is the maximum possible cardinality of according to the previous theorem. Consequently, the upper bound of given in Theorem 2.7 is indeed sharp.

3. Constructing the Graph with Maximum Cardinality of the Larger Partite Set

For a bipartite diametrical graph of diameter and partite sets and with , an obvious lower bound of is . An -graph of diameter and partite sets with equal cardinalities was constructed in [8] as follows.

Lemma 3.1. For any integer , the graph whose vertex set is the disjoint union , where and , and edge set is is an -graph of diameter .

We will denote the graph in Lemma 3.1 by . The graph is depicted in Figure 2(a).

fig1
Figure 2: Two nonisomorphic -graphs with minimum . Any two vertices of at distance 2 from each other have exactly 2 common neighbors.

The next theorem provides an example of a bipartite diametrical graph of diameter and partite sets and with in which the maximum possible value of given in Theorem 2.7 is attained.

Theorem 3.2. Suppose that is an integer. There is a unique -graph of diameter with partite sets , satisfying and .

Proof. Let and . By Lemma 2.6 and the proof of Theorem 2.7 there is only one way to make an -graph of diameter with partite sets , and with and being buddies, : put the into one-to-one correspondence with the different sets in which is one of , for each , and define adjacency by setting if corresponds to in the one-to-one correspondence. It is straightforward to see that the graph so defined is an -graph of diameter for each in the buddy of is the vertex in whose set of neighbors is

We will denote the graph constructed in the proof of Theorem 3.2 by .

The following fact follows immediately from the constructions of the graphs and .

Corollary 3.3. For any integer , if is a bipartite diametrical graph of diameter and partite sets and where , then where both inequalities are sharp.

Example 3.4. For , the graph which we have already constructed is indeed the Rhombic Dodecahedron depicted in Figure 1.

4. Some Properties of the Graphs and

We have seen in the previous section that for any integer and partite sets and with , the graph is a bipartite diametrical graph of diameter and such partite sets with minimum possible order, while is one with maximum possible order.

Now we will show that the graph is always included within the graph , recall that a subgraph of is isometric if for all , see [16].

Theorem 4.1. For any integer , the graph has an isometric subgraph isomorphic to .

Proof. Let and be the partite sets of the graph . Then, by the construction of , for each , there exists a unique vertex such that , and hence . Then,the set of vertices induces a subgraph of isomorphic to . Obviously, the buddy of a vertex of in is precisely its buddy in . To show that is isometric, let and be any two distinct vertices of . If or , then . So suppose that and . Then it is clear that when both and belong to the same partite set of . Thus let and be in different partite sets of , say and . Then for otherwise we would have which contradicts the fact that . Then, . Therefore is an isometric subgraph of

The Rhombic Dodecahedron and the even cycles are minimal diametrical graphs, while the graph is not, for if we remove an edge corresponding to , then the obtained graph is again diametrical of the same diameter. Notice also that the graph is just , which means that need not be minimal. But the graph is minimal diametrical as the next result says.

Theorem 4.2. For any integer , if is an -graph of diameter and partite sets of cardinalities and then is a minimal diametrical graph.

Proof. By Theorem 3.2, the graph is isomorphic to . Let be an edge of , where and . Then , and . Let be the vertex whose neighborhood in is . Then . Hence, . But is even, so . Clearly since Therefore, and are two distinct vertices with and , which implies that is not diametrical of diameter . Since was an arbitrary edge of the graph which is isomorphic to is minimal.

Now we can improve Theorem 3.2and show that the graph is not only the unique (up to isomorphism) -graph of diameter 4 and partite sets of cardinalities and , but also the unique (up to isomorphism) bipartite diametrical graph of such a diameter and partite sets.

Theorem 4.3. For any integer , if is a bipartite diametrical graph of diameter 4 and partite sets of cardinalities and , then is isomorphic to .

Proof. Let be a bipartite diametrical graph of diameter and partite sets and with and . In view of Theorem 3.2, it would be enough to show that is symmetric. So, assume to the contrary that is not symmetric. Then, by Theorem 2.3, the graph is a proper subgraph of an -graph of the same diameter and partite sets as . Let . Then , and hence for any we have . Therefore, is diametrical of the same diameter and partite sets as and , which contradicts the minimality of implied by the previous theorem.

The following characterization of the Rhombic Dodecahedron is a direct consequence of Theorem 4.3 and Theorem 2.7.

Corollary 4.4. The Rhombic Dodecahedron is the unique (up to isomorphism) bipartite diametrical graph of diameter 4 and partite sets and where .

According to [12], if is an -graph of order and diameter 4 with partite sets and such that , then . Moreover, if , then . It is also shown in [12] that is the unique (up to isomorphism) symmetric diametrical graph of order 12 and diameter 4 and hence in particular the unique -graph of such an order and diameter. The previous corollary shows that the Rhombic Dodecahedron is not only the unique (up to isomorphism) -graph of order 14 and diameter 4, but also the unique (up to isomorphism) bipartite diametrical graph of such an order and diameter. Combining these results with Lemma 2.4, we can completely characterize the -graphs of diameter 4 and cardinality of the smaller partite set not exceeding .

Corollary 4.5. Let be an -graph of diameter 4 and cardinality of the smaller partite set not exceeding Then is isomorphic to one of the graphs: , and the Rhombic Dodecahedron.

In the set of bipartite diametrical graphs of diameter and partite sets and with , although the graph is the unique (up to isomorphism) with maximum order, the graph needs not be the only one with minimum order, for example, the graphs and , where is an edge correspoding to , both have minimum order. Even under the condition of being symmetric, although was proven in [12] to be the only one, in general needs not be unique, for example, and the graph of diameter 4 in Figure 2 are nonisomorphic -graphs with minimum order.

Now we will prove that the set of all neighborhoods of vertices of the larger partite set in the graph forms a matroid on the smaller partite set, and determine its basis graph.

Theorem 4.6. Let and be the partite sets of with cardinalities and , respectively. Then forms a matroid on whose basis graph is the hypercube .

Proof. Let be two elements of with . Then . Thus the set has exactly elements and contains no pair of diametrical vertices. So by the construction of , there exists a vertex such that . Therefore, and hence is a matroid. Now to determine the basis graph of let be a base of the matroid obtained from a base by a single exchange. Then for some and some . But the set contains exactly one of . Then must be . Therefore, a single exchange is precisely replacing a vertex by its buddy, which means that two vertices in are adjacent if and only if one of them can be obtained from the other by replacing a vertex from by its buddy. Now since , we can represent each vertex of by a binary string where for we put thus two vertices of are adjacent if and only if their binary representations differ at exactly one place. This is just the binary representation of the hypercube .

It is shown in [8] that has exactly automorphisms. This indicates that has a lot of symmetry. The next result reflects the higher symmetry possesses.

Theorem 4.7. For any integer , the automorphism group has order .

Proof. Let be an automorphism of . Then, since preserves degrees of vertices and , it follows by Lemma 2.6 that the vertex must belong to . Then and hence since preserves distances between vertices, we must have for every vertex . Now for any vertex , by the construction of , there exists a unique vertex such that and hence . This gives an automorphism of . Thus an automorphism of is completely defined by specifying the image of every vertex in the closed neighborhood . From the proof above, the image of can be any vertex from , and the set of images of the vertices from must only be the set of neighbors of . This implies that we have exactly ways to choose and then ways to choose the set of images of the vertices from . Therefore, has exactly automorphisms.

One can easily verify that the -graphs of diameter 4 and order at most 14, already characterized in Corollary 4.5, are bipartite self complementary. The following result assures that this is the case for any -graph of diameter 4.

Theorem 4.8. If is an -graph of diameter 4, then is bipartite self complementary.

Proof. Let and be the partite sets of which is a subgraph of . Then, by Lemma 2.6, each vertex from is adjacent to exactly half of the vertices of , and is adjacent to the other half. Thus, the one-to-one correspondence from to , which fixes every vertex in and sends every vertex in to its buddy, preserves adjacency.

Consequently, both and are bipartite self complementary.

Finally, we close this section by the following result including some properties of and concerning girth, domination number, and being Eulerian.

Theorem 4.9. (i) The girth of each of the graphs and where is 4; while the graph has girth 8.
(ii) The domination number of any -graph of diameter 4 is 4. Hence, each of and has domination number 4.
(iii) Any -graph of diameter 4 is Eulerian if and only if the cardinality of each of its partite sets is divisible by 4. Hence, the graphs and are Eulerian if and only if is even.

Proof. (i) The graph is the 8-cycle. Now consider . In the graph we have and . Then is a 4-cycle in the bipartite graph which implies that the girth of is 4. Now since is bipartite, then by Theorem 4.1, its girth is also 4.
(ii) Let be an -graph of diameter 4 and partite sets and . Then by Lemma 2.6, each vertex in one of the partite sets is adjacent to exactly half of the vertices of the other partite set. Thus, cannot have a dominating set with number of elements less than 4. But for any two vertices where and , we have and . Therefore, is a dominating set of .
(iii) Let be an -graph of diameter 4 and partite sets and . By Lemma 2.6, each vertex of has degree either or . Therefore, the degree of every vertex of is even if and only if both and are multiples of 4.

One might put for further research the following question. For an -graph of diameter 4 and partite sets and where , a sharp lower bound and a sharp upper bound of are and , respectively. If then for each integer with , what about the existence of an -graph of diameter 4 and partite sets and where and ?

Acknowledgment

The authors are grateful to the referees for their valuable suggestions.

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