Antipodal Graph Research Articles

On Characterization of Balance and Consistency Preserving d-Antipodal Signed Graphs

A signed graph is an ordered pair Σ=(G,σ), where G is a graph and σ:E(G)⟶{+1,−1} is a mapping. For e∈E(G), σ(e) is called the sign of e and for any sub-graph H of G, σ(H)=∏e∈E(H)σ(e) is called the sign of H. A signed graph having a sign of each cycle +1 is called balanced. Two vertices in a graph G are called antipodal if dG(u,v)=diam(G). The antipodal graph A(G) of a graph G is the graph with a vertex set that is the same as that of G, and two vertices u,v in A(G) are adjacent if u,v are antipodal. By the d-antipodal graph GdA of a graph G, we refer to the union of G and A(G). Given a signed graph Σ=(G,σ), the signed graph ΣdA=(GdA,σd) is called the d-antipodal signed graph of G, where σd is defined as follows: σd(e)=σ(e)ife∈E(G)andotherwise,σd(e)=∏P∈Peσ(P), where Pe is the collection of all diametric paths in Σ connecting the end vertices of an antipodal edge e in ΣdA. In this article, the balance property and canonical consistency of d-antipodal signed graphs of Smith signed graphs (connected graphs having a highest eigenvalue of 2) are studied.

On the Steiner antipodal number of graphs

The Steiner n -antipodal graph of a graph G on p vertices, denoted by S A n ( G ), has the same vertex set as G and any n (2 ≤ n ≤ p ) vertices are mutually adjacent in S A n ( G ) if and only if they are n -antipodal in G . When G is disconnected, any n vertices are mutually adjacent in S A n ( G ) if not all of them are in the same component. S A n ( G ) coincides with the antipodal graph A ( G ) when n = 2 . The least positive integer n such that S A n ( G ) ≅ H , for a pair of graphs G and H on p vertices, is called the Steiner A -completion number of G over H . When H = K p , the Steiner A -completion number of G over H is called the Steiner antipodal number of G . In this article, we obtain the Steiner antipodal number of some families of graphs and for any tree. For every positive integer k , there exists a tree having Steiner antipodal number k and there exists a unicyclic graph having Steiner antipodal number k . Also we show that the notion of the Steiner antipodal number of graphs is independent of the Steiner radial number, the domination number and the chromatic number of graphs.

Some degree and distance-based invariants of wreath products of graphs

The wreath product of graphs is a graph composition inspired by the notion of wreath product of groups, with interesting connections with Geometric Group Theory and Probability. This paper is devoted to the description of some degree and distance-based invariants, of large interest in Chemical Graph Theory, for a wreath product of graphs. An explicit formula is obtained for the Zagreb indices, in terms of the Zagreb indices of the factor graphs. A detailed analysis of distances in a wreath product is performed, allowing to describe the antipodal graph and to provide a formula for the Wiener index. Finally, a formula for the Szeged index is obtained. Several explicit examples are given.

On the Automorphism Group of an Antipodal Tight Graph of Diameter 4 with Parameters (5, 7, r)

On the Automorphism Group of an Antipodal Tight Graph of Diameter 4 with Parameters (5, 7, r)

Antipodal graphs of diameter 4

The metric space (X, d) is called antipodal if for an arbitrary point x exists the one point y such that for an arbitrary point z of the set X the equality d(z, x) + d(z, y) = d(x, y) holds. In other words, the metric space (X, d) is called antipodal if for an arbitrary x there exists an antipode. For any connected undirected finite graph G we define the associated graph distance. The distance between two vertices v1 and v2 in G equals to the length of the shortest path between v1 and v2. A connected undirected finite graph G is called antipodal if its associated graph metric is antipodal. Hamming’s graphs, Johnson’s graphs and Coctail-party graphs are well-known constructions of antipodal graphs. In particular, the antipodal graphs of diameter 2 only are Cocktail-party graphs. Cocktail-party graph is the graph consisting of two rows of paired vertices in which all vertices but the paired ones are connected with a graph edge. In the article was characterized antipodal graphs of diameter 3. In this paper was showed that almost every graph is an induced subgraph of some antipodal graph P(G) of diameter 3. The construction P(G) used connected undirected finite semi-canonical graphs. Using an idea by Dragan Stevanovic from, we introduce the construction of graphs which allows to construct antipodal graphs of diameter 4. The basis of this construction is using semi-canonical graphs on a set of four vertices. A graph G is called semi-canonical if for any vertex v of this graph there is at least one vertex u of graph G, such that v and u are not connected by edge. There are only two semi-canonical graphs in a set of four vertices that are C4 and L4 the cycle and the path on 4 vertices correspondingly. So, we construct two antipodal graphs of diameter 4. We prove the antipodal of both of these graphs.

On automorphisms of distance-regular graphs with intersection arrays {2r + 1, 2r − 2, 1; 1, 2, 2r + 1}

Let Γ be an antipodal graph with intersection array {2r+1, 2r−2, 1; 1, 2, 2r+1}, where 2r(r + 1) ≤ 4096. If 2r + 1 is a prime power, then Mathon’s scheme provides the existence of an arc-transitive graph with this intersection array. Note that 2r + 1 is not a prime power only for r ∈ {7, 17, 19, 22, 25, 27, 31, 32, 37, 38, 42, 43}. We study automorphisms of hypothetical distance-regular graphs with the specified values of r. The cases r ∈ {7, 17, 19} were considered earlier. We prove that, if Γ is a vertex-symmetric graph with intersection array {2r + 1, 2r − 2, 1; 1, 2, 2r +1}, 2r + 1 is not a prime power, and r ≤ 43, then r = 25, 27, or 31.

On graphs whose local subgraphs are strongly regular with parameters (115, 18, 1, 3)

We consider undirected graphs without loops or multiple edges. If a and b are vertices in a graph Γ, then d(a, b) denotes the distance between a and b, and Γi(a) denotes the subgraph of Γ induced by the set of vertices of Γ that are a distance of i away from a. The subgraph Γ1(a) is called the neighborhood of a and is denoted by (a). Γ is called a regular graph of degree k if (a) contains precisely k vertices for any vertex a in Γ. Γ is called an edgeregular graph with parameters ( v, k, λ) if Γ is a regular graph of degree k on v vertices and each edge in Γ lies in λ triangles. Γ is said to be an amply regular graph with parameters (v, k, λ, μ) if Γ is an edgereg� ular graph with the corresponding parameters and the subgraph (a) ∩ (b) contains μ vertices in the case of d(a, b) = 2. An amply regular graph of diameter 2 is called a strongly regular graph. A graph Γ of diameter d is said to be antipodal if the relation of coincidence or being a distance of d apart on its vertex set is an equivalence relation. An antipo� dal quotient Γ' is a graph whose vertices are the antip� odal classes of Γ and two classes are adjacent if a vertex of one class is adjacent to a vertex of the other class. An antipodal graph Γ is called an rcovering (of its antip� odal quotient) if each of its antipodal classes contains precisely r vertices. If vertices u and w are separated by a distance of i in Γ, then bi(u, w) (ci(u, w)) denotes the number of vertices in the intersection of Γi +1 (u) (Γi -1 (u)) with (w). A graph Γ of diameter d is called a distanceregular graph with the intersection array {b 0 , b 1 , …, b d -1 ; c 1 , c2, …, cd} if the values bi(u, w) and ci(u, w) are indepen� dent of the choice of the vertices u and w separated by a distance of i. Let ai = k - bi - ci.

On the tight span of an antipodal graph

The tight span of a finite metric space is essentially the ‘smallest’ path geodesic space into which the metric space embeds isometrically. In this situation, the tight span is also contractible and has a natural cell structure, so that it lends itself naturally to the study of the Cayley graph of a group. As a first step in this study, we consider the tight span of a metric space which arises from the graph metric of an antipodal graph. In particular, we develop techniques for the study of the tight span of a graph, which we then apply to antipodal graphs. In this way, we are able to find the polytopal structure of the tight span for special examples of antipodal graphs. Finally, we present computer generated examples of tight spans which were made possible by the techniques developed in this paper.

The Signed Chromatic Number of the Projective Plane and Klein Bottle and Antipodal Graph Coloring

A graph with signed edges (a signed graph) is k-colorable if its vertices can be colored using only the colors 0, ±1, ..., ± k so that the colors of the endpoints of a positive edge are unequal while those of a negative edge are not negatives of each other. Consider the signed graphs without positive loops that embed in the Klein bottle so that a closed walk preserves orientation iff its sign product is positive. All of them are 2-colorable but not all are 1-colorable, not even if one restricts to the signed graphs that embed in the projective plane. If the color 0 is excluded, all are 3-colorable but, even restricting to the projective plane, not necessarily 2-colorable.

On transmitting delay in a distance-transitive strongly antipodal graph

We show that the minimum transmitting delay of a given message from a certain sender via two links to all the processors interconnected through a network of a distance-transitive, strongly antipodal graph is 1 + ⌈ D 2 ;2⌉ , where D is the diameter of the graph.

Antipodal graphs and digraphs

The antipodal graph of a graphG, denoted byA(G), has the same vertex set asGwith an edge joining verticesuandvifd(u,v)is equal to the diameter ofG. (IfGis disconnected, thendiam G=∞.) This definition is extended to a digraphDwhere the arc(u,v)is included inA(D)ifd(u,v)is the diameter ofD. It is shown that a digraphDis an antipodal digraph if and only ifDis the antipodal digraph of its complement. This generalizes a known characterization for antipodal graphs and provides an improved proof. Examples and properties of antipodal digraphs are given. A digraphDis self-antipodal ifA(D)is isomorphic toD. Several characteristics of a self-antipodal digraphDare given including sharp upper and lower bounds on the size ofD. Similar results are given for self-antipodal graphs.

A note on ‘on antipodal graphs’

A lemma stated in our earlier paper has been corrected without affecting the major content of that paper. An independent proof for the succeeding lemma in that paper has also been produced. It is be noted that given a graph G, the antipodal graph A( G) is uniquely defined, but given A( G), there exist more than one graph H such that A( H) = A( G).

Antipodal graphs of diameter three

Distance-regular graphs of diameter three are of three (almost distinct) kinds: primitive, bipartite, and antipodal. An antipodal graph of diameter three is just an r-fold covering of a complete graph K k+1 for some r⩽ k. Its intersection array and spectrum are determined by the parameters r, k together with the number c of 2-arcs joining any two vertices at distance two. Most such graphs have girth three. In this note we consider antipodal distance-regular graphs of diameter three and girth ⩾ 4. If r=2, then the only graphs are “ K k+1, k+1 minus a 1-factor.” We therefore assume r⩾3. The graphs with c=1 necessarily have r= k and were classified in lsqb3rsqb. We prove the inequality r−2>c 1 2 (Theorem 2), list the feasible parameter sets when c=2 or 3 (Corollary 1), and conclude that every 3-fold or 4-fold antipodal covering of a complete graph has girth three (Corollary 2).

Antipodal Embeddings of Graphs

An antipodal graph D of diameter d has the property that each vertex υ has a unique (antipodal) vertex υ at distance d from υ in D. We show that any such D has circuits of length 2d passing through antipodal pairs of vertices. The identification of antipodal vertex-pairs in D produces a quotient graph G with a double cover projection morphism p : D→G. Using the two-fold quotient map of surfaces π : S2→RP2 where the real projective plane is obtained from the sphere, we study the relation between embeddings of a planar graph in S2 and embeddings of G in RP2. In particular, our main theorem establishes that every planar antipodal graph D has an embedding in S2 such that p is a restriction of the projection π.