Bipartite Diametrical Graphs of Diameter 4 and Extreme Orders

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 2m |U| ≤ |W |, we prove that 2 is a sharp upper bound of |W | and construct an S-graph G 2m, 2 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 2m, 2 is the unique up to isomorphism bipartite diametrical graph of diameter 4 and partite sets of cardinalities 2m and 2, 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 ofG 2m, 2 form a matroid whose basis graph is the hypercubeQm. We prove that any S-graph of diameter 4 is bipartite self complementary, thus in particularG 2m, 2 . Finally, we study some additional properties ofG 2m, 2 concerning the order of its automorphism group, girth, domination number, and when being Eulerian.


Introduction
There are frequent occasions in which graphs with a lot of symmetries are required.One such family of graphs is that of S-graphs.S-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 k-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 diam G for all u, v ∈ V G , see 2 , that is, a diametrical graph G is symmetric if and only if V G I u, u for any u ∈ V G , where I u, u denotes the interval of G consisting of all vertices on shortest u, u -paths, see 5 .A symmetric diametrical graph is called an antipodal graph, and a bipartite antipodal graph is called an S-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 4. We provide a process to extend any given bipartite diametrical graph of diameter 4 to a maximal diametrical graph and hence an S-graph of the same diameter and partite sets.
For a bipartite diametrical graph of diameter  , of diameter 4 in which this maximum value of |W| is attained.In particular, when m 3, the graph G 2m, 2 m 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 |U| is a sharp lower bound of |W| which is previously known through a graph constructed in 8 , which we denote in this paper by G 2m, 2m .For m 3, it is proven in 12 that, up to isomorphism, the graph G 2m, 2m K 2 × C 6 is the unique symmetric diametrical graph of order 12 and diameter 4. For larger m, G 2m, 2m needs not be the unique symmetric diametrical bipartite graph of diameter 4. We give an example of an S-graph with the same cardinalities of partite sets as those of G 8, 8 but not isomorphic to G 8, 8 .On the other hand, we prove that, for any m ≥ 2, the graph G 2m, 2 m is not only the unique up to isomorphism S-graph of diameter 4 and partite sets of cardinalities 2m and 2 m , 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 4 and maximum order where the smaller partite set has cardinality 6. 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 S-graphs of diameter 4 and cardinality of the smaller partite set not exceeding 6.
Matroids constitute an important unifying structure which has many equivalent definitions.According to one of these definitions, a matroid on a finite set I is a collection β of subsets of I, called bases, which satisfies the exchange property: for all A, B ∈ β and a ∈ A − B, there exists b ∈ B − A such that A − {a} ∪ {b} ∈ β.It is well known that all the bases of a matroid have the same cardinality.The basis graph G β of a matroid β is the graph whose vertices are the bases of the matroid, where two vertices A, B are adjacent if they differ by a single exchange, that is, if the symmetric difference A − B ∪ B − A 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 G 2m, 2 m form a matroid whose basis graph is the hypercube Q m .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 G with partite sets U and W is called bipartite self complementary if G is a subgraph of K |U|,|W| which is isomorphic to its complement K |U|,|W| − E G with respect to the complete bipartite graph with the same partite sets, see 15 .We show that the graphs G 2m, 2 m and G 2m, 2m are bipartite self complementary and study some of their properties concerning girth, domination number, order of automorphism group, and when being Eulerian.

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 .In particular, when diam G 4, each partite set of G consists of a vertex v, its buddy v, and those vertices equidistant to v and v.
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 4 according to the following fact which was proven in 8 .

Lemma 2.2. Let G be a bipartite diametrical graph of diameter 4. Then the following are equivalent: i G is maximal in the class of bipartite diametrical graphs with the same partite sets as G;
ii G is maximal; iii G is symmetric.
The following theorem includes a process to extend any given bipartite diametrical graph of diameter 4 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 S-graph.Theorem 2.3.Let G be a bipartite diametrical graph of diameter 4 and partite sets U and W.Then, the following hold: i if u ∈ U and w ∈ W such that the set {u, u, w, w} of vertices is independent, then the graph G uw is bipartite diametrical of the same diameter and partite sets as G; ii if uw ∈ E G but u w / ∈ E G , then the graph G u w is bipartite diametrical of the same diameter and partite sets as G; iii if H is a graph obtained from G by repeated applications of (i) and (ii) until none of them is applicable, then H is an S-graph of the same diameter and partite sets as G. Proof.i Assume to the contrary that G uw is either not diametrical or has diameter less than 4. Then there exists a vertex v ∈ V G such that d G uw v, v < 4. But, since G uw is also bipartite with the same partite sets as G and both v , v belong to the same part of V G , then Thus there exists a v, v -path P in G uw containing the new edge uw and having length 2. Then either {v, v} {u, u} and P is u w u or {v, v} {w, w} and P is w u w, which gives a contradiction in both cases since wu, wu / ∈ E G uw .Therefore, G uw is a diametrical graph of diameter 4.
ii Let uw ∈ E G such that u w / ∈ E G with u ∈ U and w ∈ W, where U and W are the partite sets of G. Then u w / ∈ E G because otherwise d G u, u 2, which is not the case.Similarly, u w / ∈ E G .Now, we will show by contradiction that G u w is diametrical of diameter 4. Assume not, then there exists Clearly, G u w is bipartite with the same partite sets as G since u ∈ U and w ∈ W. But both v and v belong to the same part of V G , hence d G u w v, v is even, and so d G u w v, v 2. Thus, there exists a v, v -path P in G u w containing the new edge u w and having length 2. Consequently, either {v, v} {u, u} and P is u w u or {v, v} {w, w} and P is w u w, which gives a contradiction in both cases since uw, uw / ∈ E G u w .Therefore, G u w is bipartite diametrical of diameter 4.
iii Let H be a graph obtained from G 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 H is maximal in the class of bipartite diametrical graphs of diameter 4 and partite sets U and W. Assume to the contrary that there exist two nonadjacent vertices u ∈ U and w ∈ W of H such that H uw still diametrical of diameter 4.Then, since ii is no more applicable on H, we have u w / ∈ E H .But also i is not applicable on H, so the set {u, u, w, w} is not independent, therefore either  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 |W|, let us recall the following fact proven in 8 .

Lemma 2.6. Every vertex v of an S-graph G of diameter 4 is adjacent to exactly half of the vertices in the partite set of G not containing v.
We are now in a position to prove our improvement of the upper bound of |W|.Proof.Let U {u 1 , u * 1 , . . ., u m , u * m } and W {w 1 , . . ., w 2 m }.By Lemma 2.6 and the proof of Theorem 2.7 there is only one way to make an S-graph of diameter 4 with partite sets U, W and with u i and u * i being buddies, i 1, . . ., m: put the w j into one-to-one correspondence with the 2 m different sets S {u 1 , . . ., u m } in which u i is one of u i , u * i for each i 1, . . ., m, and define adjacency by setting N w j S if w j corresponds to S in the one-to-one correspondence.It is straightforward to see that the graph so defined is an S-graph of diameter 4; for each w in W, the buddy of w is the vertex in W whose set of neighbors is U − N w .
We will denote the graph constructed in the proof of Theorem 3.

Some properties of the graphs G 2m, 2m and G 2m, 2 m
We have seen in the previous section that for any integer m ≥ 2 and partite sets U and W with 2m |U| ≤ |W|, the graph G 2m, 2m is a bipartite diametrical graph of diameter 4 and such partite sets with minimum possible order, while G 2m, 2 m is one with maximum possible order.Now we will show that the graph G 2m, 2m is always included within the graph Proof.Let U {u 1 , u 1 , u 2 , u 2 , . . ., u m , u m } and W {w 1 , w 2 , . . ., w 2 m } be the partite sets of the graph G 2m, 2 m .Then, by the construction of G 2m, 2 m , for each i 1, 2, . . ., m, there exists a unique vertex The Rhombic Dodecahedron and the even cycles C 2n n ≥ 2 are minimal diametrical graphs, while the graph K 2 × C 6 is not, for if we remove an edge e corresponding to K 2 , then the obtained graph K 2 × C 6 − e is again diametrical of the same diameter.Notice also that the graph K 2 × C 6 is just G 6, 6 , which means that G 2m, 2m needs not be minimal.But the graph G 2m, 2 m is minimal diametrical as the next result says.Now we can improve Theorem 3.2 and show that the graph G 2m, 2 m is not only the unique up to isomorphism S-graph of diameter 4 and partite sets of cardinalities 2m and 2 m , but also the unique up to isomorphism bipartite diametrical graph of such a diameter and partite sets.a base N w i by a single exchange.Then N w j N w i − {a} ∪ {b} for some a ∈ N w i and some b / ∈ N w i .But the set N w j contains exactly one of a, a. Then b must be a.Therefore, a single exchange is precisely replacing a vertex by its buddy, which means that two vertices in G β are adjacent if and only if one of them can be obtained from the other by replacing a vertex from U by its buddy.Now since U {u 1 , u 1 , u 2 , u 2 , . . ., u m , u m }, we can represent each vertex N w of G β by a binary string r 1 r 2 N w , and hence since f preserves distances between vertices, we must have f u f u for every vertex u ∈ U − N w 1 N w 1 .Now for any vertex x ∈ W − {w 1 }, by the construction of G 2m, 2 m , there exists a unique vertex y ∈ W such that N y f N x and hence f x y.This gives an automorphism f of G 2m, 2 m .Thus an automorphism of G 2m, 2 m is completely defined by specifying the image of every vertex in the closed neighborhood N w 1 .From the proof above, the image of w 1 can be any vertex from W, and the set of images of the vertices from N w 1 must only be the set of neighbors of f w 1 .This implies that we have exactly 2 m ways to choose f w 1 and then m! ways to choose the set of images of the vertices from N w 1 .Therefore, G 2m, 2 m has exactly 2 m m! automorphisms.
One can easily verify that the S-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 S-graph of diameter 4. Proof.Let U {u 1 , u 2 , . . ., u k , u 1 , u 2 , . . ., u k } and W {w 1 , w 2 , . . ., w t , w 1 , w 2 , . . ., w t } be the partite sets of G which is a subgraph of K 2k,2t .Then, by Lemma 2.6, each vertex x from U is adjacent to exactly half of the vertices of W, and x is adjacent to the other half.Thus, the oneto-one correspondence φ from V G to V K 2k,2t − E G , which fixes every vertex in W and sends every vertex in U to its buddy, preserves adjacency.
Consequently, both G 2m, 2m and G 2m, 2 m are bipartite self complementary.Finally, we close this section by the following result including some properties of G 2m, 2m and G 2m, 2 m concerning girth, domination number, and being Eulerian.

Lemma 2 . 1 .
If G is a bipartite diametrical graph of even diameter d and partite sets U and W, then for each v ∈ V, both v and v belong to the same partite set, and hence |U| and |W| are even.

Lemma 2 . 5 .
Let u and v be distinct vertices of a diametrical graphG of diameter d ≥ 3. Then N u / ⊆ N v .Proof.If u v, then N u ∩ N v φ since d u, v ≥ 3.Suppose that u / v and assume to the contrary that N u ⊆ N v .Any shortest path joining v and u must include a vertex from N u ; let x be such a vertex in such a path.Thend v, u d v, x d x, u d v, x 1.But x ∈ N u ⊆ N v and v is the only vertex of G at distance d from v, hence d v, x d − 1. Therefore,d v, ud − 1 1 d, contradicting the uniqueness of the buddy vertex of v.

Corollary 3 . 3 .
2 by G 2m, 2 m .The following fact follows immediately from the constructions of the graphs G 2m, 2m and G 2m, 2 m .For any integer m ≥ 2, if G is a bipartite diametrical graph of diameter 4 and partite sets U and W, where 2m |U| ≤ |W|, then 2m ≤ |W| ≤ 2 m , where both inequalities are sharp.Example 3.4.For m 3, the graph G 2m, 2 m G 6 , 8 which we have already constructed is indeed the Rhombic Dodecahedron depicted in Figure 1.

Theorem 4 . 2 .
For any integer m ≥ 2, if H is an S-graph of diameter 4 and partite sets of cardinalities 2m and 2 m , then H is a minimal diametrical graph.Proof.By Theorem 3.2, the graphH is isomorphic to G 2m, 2 m .Let uw be an edge of G G 2m, 2 m , where u ∈ U and w ∈ W. Then N G−uw w N G w − {u}, N G w U − N G w , and u ∈ N G w .Let x ∈ W be the vertex whose neighborhood in G is N G x N G w − {u} ∪ {u}.Then N G−uw x ∩ N G−uw w φ.Hence, d G−uw w, x / 2. But d G−uw w, x is even, so d G−uw w, x ≥ 4.Clearly x / w since u / ∈ N G w .Therefore, x and w are two distinct vertices with d G−uw w, x ≥ 4 and d G−uw w, w ≥ 4, which implies that G − uw is not diametrical of diameter 4. Since uw was an arbitrary edge of G 2m, 2 m , the graph H which is isomorphic to G 2m, 2 m is minimal.

Theorem 4 . 8 .
If G is an S-graph of diameter 4, then G is bipartite self complementary.

Theorem 4 . 9 .
i The girth of each of the graphs G 2m, 2m and G 2m, 2 m , where m ≥ 3, is 4; while the graph G 4, 4 has girth 8.
unique diametrical vertex v, the vertex v is called the buddy of v, see 4 and 5 .A diametrical graph G is called minimal if removing any edge of G produces a graph which is not diametrical of the same diameter.A diametrical graph G in which u v ∈ E G , whenever uv ∈ E G , is called harmonic.A diametrical graph G is called maximal if there is no diametrical graph H with the same vertex set and diameter as G such that E H contains E G properly.A diametrical graph G is called symmetric if d u, v d v, u vertices, u and v, of a nontrivial connected graph G are said to be diametrical ifd u, v diam G .A nontrivial connected graph G is called diametrical if each vertex v of G has a |U| .In this paper, we improve this upper bound by proving that |W| ≤ 2 m , where m 1/2 |U|.To show that 2 m is a sharp upper bound, we construct a bipartite diametrical graph, which we denote by G 2m, 2 4 and partite sets U and W with |U| ≤ |W|, it is given in 8 that |W| ≤ |U| 1/2 uw ∈ E H or uw ∈ E H . Consequently, either d H w, w 2 or d H u, u 2, a contradiction.The only bipartite diametrical graph of diameter 4 in which the smaller partite set has cardinality less than 6 is the 8-cycle as Lemma 2.4 proven in 8 says.If G is a bipartite diametrical graph of diameter 4 and partite sets U and W with |U| ≤ |W|, then |U| ≥ 6 unless G C 8 .For a bipartite diametrical graph of diameter 4 and partite sets U and W with |U| ≤ |W|, we have |U| / 2, and whenever |U| 4, we must have |W| 4. It was shown in 8 that if |U| ≥ 6, then |W| ≤ 2m m , where m 1/2 |U|.Thus for |U| 6, we have |W| ≤ 20.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.

3. Constructing the graph G 2m, 2 m with maximum cardinality of the larger partite set For
Theorem 2.7.Let G be a bipartite diametrical graph of diameter 4 and partite sets U and W, where 2m |U| ≤ |W|.Then, |W| ≤ 2 m .Proof.By Theorem 2.3, there exists an S-graph H with the same diameter and partite sets as G which has G as a subgraph.Then, by Lemma 2.6, each vertex w ∈ W has exactly m neighbors in U. Clearly, N H w cannot contain a vertex and its buddy.But Lemma 2.5 implies that the |W| sets N H w , w ∈ W are mutually distinct.Hence, |W| must be less than or equal to the number of ways of choosing single representatives from each of m two-element sets; clearly that number is 2 m .Now, if G is a bipartite diametrical graph of diameter 4 and partite sets U and W, where |U| ≤ |W|, then by Lemma 2.1, |U| 2m for some positive integer m.Since diam G 4, we must have m ≥ 2. In the next section, we will construct for each integer m ≥ 2 a bipartite diametrical graph of diameter 4 and partite sets U and W with |U| 2m and |W| 2 m , which is the maximum possible cardinality of W according to the previous theorem.Consequently, the upper bound of |W| given in Theorem 2.7 is indeed sharp.
a bipartite diametrical graph of diameter 4 and partite sets U and W with 2m |U| ≤ |W|, an obvious lower bound of |W| is 2m.An S-graph of diameter 4 and partite sets with equal cardinalities was constructed in 8 as follows.Lemma 3.1.For any integer m ≥ 2, the graph G whose vertex set is the disjoint union U ∪ W, where U {u 1 , u 1 , u 2 , u 2 , . . ., u m , u m } and W {w 1 , w 1 , w 2 , w 2 , . . ., w m , w m }, and edge set is Then,the set of vertices U ∪ {y 1 , y 1 , y 2 , y 2 , . . ., y m , y m } induces a subgraph H of G 2m, 2 m isomorphic to G 2m, 2m .Obviously, the buddy of a vertex of H in H is precisely its buddy in G 2m, 2 m .To show that H is isometric, let x and z be any two distinct vertices of H.If x z or xz ∈ E H , then d H x , z d G 2m,2 m x , z .So suppose that x / z and xz / ∈ E H . Then it is clear that d H x , z d G 2m,2 m x , z 2 when both x and z belong to the same partite set of H. Thus let x and z be in different partite sets of H, say x ∈ U and z ∈ W H {y 1 , y 1 , y 2 , y 2 , . . ., y m , y m }.Then xz / ∈ E G 2m, 2 m for otherwise we would have N G 2m,2 m z ⊇ N H z ∪ {x} which contradicts the fact that deg G 2m,2 m z m.Then, d H x , z d G 2m,2 m x , z 3. Therefore H is an isometric subgraph of G 2m, 2 m .
• • • r m , where for i 1, 2, ..., m, we put are adjacent if and only if their binary representations differ at exactly one place.This is just the binary representation of the hypercube Q m .It is shown in 8 that G 2m, 2m has exactly 8m automorphisms.This indicates that G 2m, 2m has a lot of symmetry.The next result reflects the higher symmetry G 2m, 2 m possesses.For any integer m ≥ 3, the automorphism group Aut G 2m, 2 m has order 2 m m!.Proof.Let f be an automorphism of G 2m, 2 m .Then, since f preserves degrees of vertices and |U| / |W|, it follows by Lemma 2.6 that the vertex f w 1 w must belong to W. Then f N w 1