The antiregular connected graph on

A graph

The graphs we consider are

The disconnected graph with two components

For the linear transformation

The distinct eigenvalues

A

(i) A

Equivalently, (ii) if the monotonic nonincreasing degree sequence,

If the parts of a threshold graph partition of

An

The partition

The overall aim of this paper is to explore the spectrum of its adjacency matrix and show common properties with those of connected threshold graphs, having an equitable partition with a minimal number

The paper is organised as follows. In Section

Our main results are as follows.

In Section

In Section

We show in Section

We end with a discussion, in Section

A cograph is the union or the join of subgraphs of the form

Cographs have received much attention since the 1970s. They were discovered independently by many authors including Jung [

Connected graphs, which are

By construction, a connected cograph also has a dominating vertex. Therefore, its complement has at least one isolated vertex. A necessary condition for a connected graph to have a connected complement is that it has

Recall that

A cograph can be represented uniquely by a

Cotree

In this section we present some of the various representations of threshold graphs. Collectively, they provide a wealth of information that determine combinatorial properties of these graphs. We start with the cotree representation as in the previous section. There are certain restrictions on the structure of a cotree in the case when a cograph is a threshold graph.

We give a proof to the following result quoted in [

If a cograph

A threshold graph

Representations of

We now present various other representations of threshold graphs that are used in the proofs that follow.

A

The cotree of a threshold graph is a caterpillar.

The vertex set of a

The first vertex labelling (which we will refer to as Lab1) of a threshold graph is according to its construction. Starting from

Cotree

The Ferrers/Young diagram for a threshold graph.

In

Our labelling of the

According to our labelling convention (Lab1) for

For the purposes of inputting an

The graph of Figure

The last representation of a threshold graph that we now give is constructed from the degree sequence. Following Definition

It is well known that there exist nonisomorphic graphs with the same degree sequence. A graph determined, up to isomorphism, by its degree sequence is said to be a

A threshold graph is a unigraph.

The degree sequence

For a threshold graph,

An interesting algorithm was presented in [

The

The rows and columns of the adjacency matrix constructed in Theorem

The antiregular graph

An

We shall use the

An induced subgraph

When

The threshold graph

The connected antiregular graph

On taking the complement of

The threshold graph

The threshold graph

Let

Figures

The binary codes for the connected antiregular graphs

Since the binary code follows the construction of

The construction of connected antiregular graphs is as follows: for

The complement of a connected threshold graph

The cotree

The complement

Since

The binary string coding of the threshold graph

Similarly the binary string coding of the threshold graph

A pair of

In this section we adopt the vertex labelling Lab2 of a threshold graph induced by the Ferrers/Young diagram in accordance with the procedure to form the “stepwise” adjacency matrix presented in Theorem

A graph with duplicates is often considered as having repeated vertices and therefore redundant properties. We call the induced subgraph of a graph obtained by removing repeated vertices

An upper bound for the nullity

When the adjacency matrix

The bound in Theorem

Let

The last

A threshold graph may have duplicate vertices even if

The nullity

Let

In the proof of Theorem

If a threshold graph is singular, then no kernel eigenvector has more than two nonzero entries.

Note that any repeated rows in the first

That an antiregular graph has exactly one pair of either duplicates or coduplicates follows from its construction.

An antiregular graph

An antiregular graph

The graph

To obtain the number of duplicate and coduplicate vertices in a threshold graph, we count the number of vertices to be removed from

A threshold graph with nondegenerate representation

Most of the information to determine the grounds for a labelled graph

Let

Hence, an MC with core

Two minimal configurations:

A basis for the nullspace

Such a minimal basis for

Let

To give an example supporting Theorem

A graph of nullity two with two MCs as subgraphs.

From Theorem

The only MC found in a threshold graph as a subgraph is

Corollary

All MCs with core order at least three have

Suppose an MC is

Since

The second largest eigenvalue of

The second largest eigenvalue of an

The only MC for which the bound is known to be strict is the seven-vertex nut graph of Figure

The main eigenvalues of a graph

The polynomial

For a proof of the following result, see [

The main characteristic polynomial

Recall that

All eigenvalues of

Let Prop

Prop(2) refers to

This establishes the base cases.

Assume that Prop

Consider

The eigenvalue

If

If

By Theorem

A graph has the same number of main eigenvalues as its complement.

Let

Let

Let

If

The induction hypothesis is as follows: assume that Prop

We show that this is also true for a nondegenerate

We deduce immediately a spectral property of a threshold graph and its underlying antiregular graph.

The nondegenerate threshold graph

An equitable partition

The main part of the spectrum of

Let

We now show that the main part of the spectrum of

Let the threshold graph

The vertex labelling Lab1 is used. Let the vertices be labelled in order starting from those corresponding to

We give an example to clarify the procedure. Consider the threshold graph

For

For a graph with

The columns

The matrix

Note that

The rank of the walk matrix of

The number of walks of length

The number

Since 0 is never a main eigenvalue of

By Theorem

We note that for many threshold graphs

Now we add vertices to the degenerate form

This is also the case for some instances of the threshold graphs

If

We conclude with a note on the distribution of the eigenvalues of a threshold graph. In [

For

To transform

a vertex duplicate to the

a vertex coduplicate to the

The proof is by induction on

The spectra of the three smallest antiregular graphs,

Assume that the theorem is true for

We prove it true for

If

If, on the other hand,

We apply Lemma

The result follows by induction on

In this section, we shall represent the antiregular graph

If on adding a vertex to a graph (i) the multiplicity of an eigenvalue

We shall write

First we see an application of Lemma

Similarly, for odd

The case for even

If the threshold graph

The simple graphic appeal of the Ferrers/Young diagram

Let

This paper was supported by the Research Project Funds MATRP01-01 Graph Spectra and Fullerene Molecular Structures of the University of Malta.