The distance

The study of sequences in Graph Theory is not new. A

Next distance based sequences were the

We will now define all the terms that give graph theoretic expressions for the above discussed cases.

For all undefined terms we refer to [

Let

The distance

The sequence of numbers of vertices having degree

The

The minimum of the eccentricities is the

A graph

If

The

The distance and path degree sequences of a vertex are generalizations of the degree of a vertex. The

The

The

The

The status of a vertex is also called the

The

As an illustration consider the graph

If we consider the 3-dimensional cube

In [

Between pairs of atoms in acyclic structures there is a unique path so that the number of paths of a given length corresponds to the number of neighbors at a given distance.

This can be rephrased in graph theoretic terms as follows.

For a connected graph

A nonassertive proof was given by Quintas and Slater [

Hence, this kind of study leads to the problem of finding the smallest order for which there exists a pair of nonisomorphic graphs which are equivalent relative to some set of graphical invariants. So the following conjecture was made in line with the above conjecture by Randic [

Two graphs

This conjuncture was also shown to be invalid, by citing infinite class of pairs of nonisomorphic trees having the property that the two members of each pair have the same path degree sequence by Quintas and Slater [

As noted above, a tree is not, in general, characterized by its path (or distance) degree sequence. The least order for which there exists a pair of nonisomorphic trees with the same path degree sequence has been shown to be more than

On similar lines one can search for pairs of nonisomorphic graphs having the same distance degree sequence. Slater [

A pair of nonisomorphic trees with the same path degree sequence.

Smallest order graphs with the same distance degree sequence.

And this is the only such pair as noted by Quintas and Slater in [

If

Two problems were asked in the same paper.

What is the smallest order

What is the smallest order

With reference to Problem

If

If

If

For several values of

If one asks for the smallest order for which the distance degree sequence fails to distinguish between

Let

As noted above, there does not exist a pair of nonisomorphic graphs with the same path degree sequence on less than

For

In fact, a paper by Slater [

In [

its description by means of its edges and for

the coefficients of its characteristic polynomial;

the eigenvalues of its

the number of its cycles of length

its diameter, connectivity, and planarity;

the order of its automorphism group.

From their table the girth, the circumference, and the chromatic number are easily determined. The graphs were classified according to their order and within each such class the graphs were ordered lexicographically according to their eigenvalues, which for each graph were listed in nonincreasing order. In addition, a number of observations concerning the spectral properties of these graphs were made. The study was motivated by the importance of cubic graphs and by the search for cospectral cubic graphs. The cubic graph generation is looked into by Brinkmann [

its distance degree sequence;

its distance distribution;

the mean distance at each vertex;

the mean distance for the graph;

its path degree sequence;

the number of paths of specified length;

the total number of paths for the graph.

For each graph a number of other parameters can be determined from these tables. The radius, the diameter, and the eccentricity of every vertex can be read directly by noting the length of the appropriate distance degree sequence. By inspection one can determine the order of the center and of the median of the graph. In addition, the computation of other parameters is facilitated by making use of the given data.

Now we have the stage set for the discussion on the highly regular class of graphs involving the distance degree sequences, namely, the

For example, the three-dimensional cube

In [

Each regular graph of diameter ≤2 is DDR and the complement of each regular graph of diameter ≥3 is DDR.

This result ensures that “every regular graph with diameter at most two is DDR.” We know that every DDR graph is regular, but the distribution of DDR graphs in regular graphs is not clear. So, the following question is relevant.

What proportion of

Following constructions help in getting DDR graphs of arbitrarily chosen diameter and degree. For these methods we require the following terms.

Let

Note that the special case

So the next result gives a family of DDR graphs.

If

This result can be extended to

The existence of a DDR graph having diameter

There exist DDR graphs having diameter

Observe that the only connected

But, characterizing DDR graphs of higher diameter is challenging. In [

Characterize DDR graphs having diameter ≥3.

Since then there is no answer to this question. A step forward was taken by Huilgol et al. [

Let

The proof of the above theorem is primarily based on the following two results.

Every connected self-centered graph of order

In a connected graph

As the diameter three DDR graphs are considered, which are self-centered graphs, satisfying the inequality,

For any integer

So, the construction of new families of DDR graphs is very interesting. In [

The lexicographic product is defined as follows.

Given graphs

Let

This characterizes the generalized lexicographic products. Next result uses the generalized lexicographic product to show the existence of a DDR graph of arbitrary diameter.

There exists a DDR graph of diameter

There exists a DDR graph

If

In [

And the normal product of two graphs

Cartesian product of two graphs

Let

The next result characterizes the normal product of two DDR graphs.

Normal product

As mentioned earlier, the relation of DDR graphs with other graph properties is relatively an unexplored area. In this section we try to relate DDR graphs with some symmetry properties.

Let

There are natural numbers

the number of vertices at distance

the number of vertices at distance

In [

A graph that is vertex-transitive or distance-regular must be DDR.

In the next result Bloom et al. [

Distance degree regularity does not imply edge-transitivity, vertex-transitivity, nor distance-regularity. Also, an edge-transitive graph is not necessarily DDR (and hence neither vertex-transitive, symmetric, nor distance transitive).

Vertex-transitivity does not imply edge-transitivity, symmetry, nor distance-regularity. Using the following Bloom et al. [

The graph shown in Figure

Distance-regularity does not imply vertex-transitivity. This is shown by an example, noted by Biggs, in [

Symmetry implies neither distance-transitivity nor distance-regularity. The former observation is demonstrated by a DDR example due to Frucht et al. [

Bloom et al. [

Which

It is interesting to know that although most symmetry conditions suffice to make a graph DDR, there are DDR examples among the least symmetric graphs. One such class is DDR graphs having identity automorphism group. If

If

For graphs with regularities 3 and 4 the situation is not clear. Although a least-order

If

Next we will consider the relation of DDR graphs with a binary relation defined on the graph, namely, the eccentric digraph. The

In [

There exists a DDR graph

There exists a DDR graph

For a unique eccentric vertex DDR graph, if

For a given diameter

The unique eccentric vertex DDR graphs have all their iterated eccentric digraphs as DDR graphs.

For nonunique eccentric vertex DDR graphs the problem remains still open.

When do nonunique eccentric vertex DDR graphs have all their iterated digraphs as DDR graphs?

In contrast to distance degree regular (DDR) graphs the Distance Degree Injective (DDI) graphs are the graphs with no two vertices of

If

The converse is obvious as there are DDR graphs with identity automorphism group.

As DDR graphs, the characterizations elude DDI graphs. But there are many existential results and examples of DDI graphs.

A smallest order nontrivial DDI graph has order 7 and there exist DDI graphs having order

The smallest order nontrivial identity graph has 6 vertices and there are exactly eight identity graphs on 6 vertices and these are not DDI. So the least order is 7. Actually, the identity tree as shown in Figure

The following result shows the existence of a DDI graph of arbitrary diameter.

A smallest diameter nontrivial DDI graph has diameter

The graphs shown in Figure

If

The following theorem is useful in showing that a graph is not DDI.

If

Note that the complement of the diameter three graph shown in Figure

But the picture is not clear when we wish to have a graph and its complement to be DDI.

Does there exist a graph

So the next question was on

Does there exist a nontrivial

This problem got resolved by the existence of a cubic DDI graph on 24 vertices and having diameter 10 as found in [

Let

Since the distance degree sequence of a graph is independent of a labeling, this result shows that almost all

In this direction there was one more problem defined by Halberstam and Quintas [

For

Martinez and Quintas [

From [

if there is a cubic DDI graph having less than 18 vertices, then its order must be 16;

if there is a cubic DDI graph having diameter less than 7, then its diameter must be 4, 5, or 6.

So all these cases were considered by Huilgol and Rajeshwari [

There does not exist a cubic DDI graph of order 16 with diameters 4, 5, and 6.

So the graph of order 18 as in [

On the other extreme, construction of new DDI graphs from the smaller order/diameter is also interesting and is looked into by Huilgol et al. [

If the cartesian product of two graphs

But the converse is not true, as the cartesian product of the DDI graph represented by Figure

Let

Let

Let

Similar results are proved for normal product of DDI graphs.

If the normal product

As in the case of cartesian product, the normal product of two DDI graphs need not be DDI and the same example serves the purpose.

Let

Let

Let there be two sets

Kennedy and Quintas [

Does there exist a pair of nonisomorphic graphs having the same distance degree sequence and such that one of them is nonplanar with a subgraph homeomorphic to

Does there exist a pair of nonisomorphic graphs having the same path degree sequence and such that only one of the graphs is planar?

At the other extreme Huilgol et al. [

Any graph can be embedded in a DDR graph.

DDR graphs exhibit high regularity in terms of the vertices and their distance distribution. If we relax one vertex to have different distance degree sequence, then we call such a graph an almost distance degree regular graph, or in short, an ADDR graph. Similarly, we define ADDI graphs.

A graph

A graph

Embedding in DDI graphs seems not so easy. But Huilgol et al. [

Any path can be embedded in an almost DDI graph.

Every cycle can be embedded in a DDI graph.

So the following problems were posed in [

Can any graph be embedded in a DDI graph?

Does there exist a

One more problem can be posed at this juncture as follows.

Does there exist a self-centered,

We conclude this article with a comment that even though there are numerous examples and results on DDR and DDI graphs available in literature, the characterizations elude. Hence, many open problems in these areas still persist and keep researchers working.

The author declares that there is no conflict of interests regarding the publication of this paper.