Distance Degree Regular Graphs and Distance Degree Injective Graphs: An Overview

The distance d ( v , u ) from a vertex v of G to a vertex u is the length of shortest v to u path. The eccentricity e v of v is the distance to a farthest vertex from v . If d ( v , u ) = e ( v ) , ( u ≠ v ) , we say that u is an eccentric vertex of v . The radius rad ( G ) is the minimum eccentricity of the vertices, whereas the diameter diam ( G ) is the maximum eccentricity. A vertex v is a central vertex if e ( v ) = r a d ( G ) , and a vertex is a peripheral vertex if e ( v ) = d i a m ( G ) . A graph is self-centered if every vertex has the same eccentricity; that is, r a d ( G ) = d i a m ( G ) . The distance degree sequence (dds) of a vertex v in a graph G = ( V , E ) is a list of the number of vertices at distance 1 , 2 , . . . . , e ( v ) in that order, where e ( v ) denotes the eccentricity of v in G . Thus, the sequence ( d i 0 , d i 1 , d i 2 , … , d i j , … ) is the distance degree sequence of the vertex v i in G where d i j denotes the number of vertices at distance j from v i . The concept of distance degree regular (DDR) graphs was introduced by Bloom et al., as the graphs for which all vertices have the same distance degree sequence. By definition, a DDR graph must be a regular graph, but a regular graph may not be DDR. A graph is distance degree injective (DDI) graph if no two vertices have the same distance degree sequence. DDI graphs are highly irregular, in comparison with the DDR graphs. In this paper we present an exhaustive review of the two concepts of DDR and DDI graphs. The paper starts with an insight into all distance related sequences and their applications. All the related open problems are listed

Products of distance degree regular and distance degree injective graphs.

The eccentricity e (u) of a vertex u is the maximum distance of u to any other vertex in G. The distance degree sequence (dds) of a vertex v in a graph G = (V, E) is a list of the number of vertices at distance 1, 2, …, e (u) in that order, where e (u) denotes the eccentricity of u in G. Thus the sequence is the dds of the vertex vi in G where denotes number of vertices at distance j from Vi . A graph is distance degree regular (DDR) graph if all vertices have the same dds. A graph is distance degree injective (DDI) graph if no two vertices have same dds. In this paper we consider Cartesian and normal products of DDR and DDI graphs. Some structural results have been obtained along with some characterizations

Embedding in Distance Degree Regular and Distance Degree Injective graphs

The eccentricity e(u) of a vertex u is the maximum distance of u to any other vertex of G.The distance degree sequence (dds) of a vertex u in a graph G = (V, E) is a list of the number of vertices at distance 1, 2,. . . , e(u) in that order, where e(u) denotes the eccentricity of u in G. Thus the sequence (di0 , di1 , di2 , . . . , dij , . . .) is the dds of the vertex vi in G where dij denotes number of vertices at distance j from vi . A graph is distance degree regular (DDR) graph if all vertices have the same dds. A graph is distance degree injective (DDI) graph if no two vertices have the same dds. In this paper, we consider the construction of a DDR graph having any given graph G as its induced subgraph. Also we consider construction of some special class of DDI graphs. Keywords: Distance degree sequence, Distance degree regular (DDR) graphs, Almost DDR graphs, Distance degree injective(DDI) grap

On Distance Magic Harary Graphs

This paper establishes two techniques to construct larger distance magic and (a, d)-distance antimagic graphs using Harary graphs and provides a solution to the existence of distance magicness of legicographic product and direct product of G with C4, for every non-regular distance magic graph G with maximum degree |V(G)|-1.Comment: 12 pages, 1 figur

Distance Degree Regular Graphs and Theireccentric Digraphs

The eccentricity e(u) of a vertex u is the maximum distance of u to any other vertex of G.The distance degree sequence (dds) of a vertex v in a graph G = (V,E) is a list of the number of vertices at distance 1, 2, . . . , e(u) in that order, where e(u) denotes the eccentricity of v in G. Thus the sequence (di0 , di1 , di2 , . . . , dij , . . .) is the dds of the vertex vi in G where dij denotes number of vertices at distance j from vi. A graph is distance degree regular (DDR) graph if all vertices have the same dds. A vertex v is an eccentric vertex of vertex u if the distance from u to v is equal to e(u). The eccentric digraph ED(G) of a graph (digraph) G is the digraph that has the same vertex as G and an arc from u to v exists in ED(G) if and only if v is an eccentric vertex of u in G. In this paper, we consider the construction of new families of DDR graphs with arbitrary diameter. Also we consider some special class of DDR graphs in relation with eccentric digraph of a graph. Different structural properties of eccentric digraphs of DDR graphs are dealt herewith

A short proof of the odd-girth theorem

Recently, it has been shown that a connected graph $\Gamma$ with $d+1$ distinct eigenvalues and odd-girth $2d+1$ is distance-regular. The proof of this result was based on the spectral excess theorem. In this note we present an alternative and more direct proof which does not rely on the spectral excess theorem, but on a known characterization of distance-regular graphs in terms of the predistance polynomial of degree $d$

Asymptotic Delsarte cliques in distance-regular graphs

We give a new bound on the parameter $\lambda$ (number of common neighbors of a pair of adjacent vertices) in a distance-regular graph $G$, improving and generalizing bounds for strongly regular graphs by Spielman (1996) and Pyber (2014). The new bound is one of the ingredients of recent progress on the complexity of testing isomorphism of strongly regular graphs (Babai, Chen, Sun, Teng, Wilmes 2013). The proof is based on a clique geometry found by Metsch (1991) under certain constraints on the parameters. We also give a simplified proof of the following asymptotic consequence of Metsch's result: if $k\mu = o(\lambda^2)$ then each edge of $G$ belongs to a unique maximal clique of size asymptotically equal to $\lambda$, and all other cliques have size $o(\lambda)$. Here $k$ denotes the degree and $\mu$ the number of common neighbors of a pair of vertices at distance 2. We point out that Metsch's cliques are "asymptotically Delsarte" when $k\mu = o(\lambda^2)$, so families of distance-regular graphs with parameters satisfying $k\mu = o(\lambda^2)$ are "asymptotically Delsarte-geometric."Comment: 10 page

Distinct Distances in Graph Drawings

The \emph{distance-number} of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the distance-number of trees, graphs with no $K^-_4$-minor, complete bipartite graphs, complete graphs, and cartesian products. Our main results concern the distance-number of graphs with bounded degree. We prove that $n$-vertex graphs with bounded maximum degree and bounded treewidth have distance-number in $\mathcal{O}(\log n)$. To conclude such a logarithmic upper bound, both the degree and the treewidth need to be bounded. In particular, we construct graphs with treewidth 2 and polynomial distance-number. Similarly, we prove that there exist graphs with maximum degree 5 and arbitrarily large distance-number. Moreover, as $\Delta$ increases the existential lower bound on the distance-number of $\Delta$-regular graphs tends to $\Omega(n^{0.864138})$