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The base set of primitive zero-symmetric sign pattern matrices with zero diagonal is

A sign pattern matrix (or sign pattern)

From now on, we assume that all the matrix operations considered in this paper are operations of the matrices over the set

In [

Let

For a sign pattern matrix

A nonnegative square matrix

A square sign pattern matrix

It is well known that graph-theoretical methods are often useful in the study of the powers of square matrices, so we now introduce some graph-theoretical concepts.

Let

Let

Let

It is well known that a digraph

Let

We say that a sign pattern matrix

The base set of primitive ZS sign pattern matrices and the base set of primitive ZS sign pattern matrices with zero diagonal are given, respectively, in [

In this paper, we characterize the primitive sign pattern matrices with zero diagonal attaining the maximum base. Our main result is given in the following theorem.

Let

The graph

The proof of Theorem

In this section, we introduce some theorems, definitions, and lemmas which we need to use in the proof of our main result in Section

In [

Let

Let

By Theorems

Two walks

Let

If

both

Let

There is an integer

If there exists a pair of SSSD walks of length

The minimal such

Suppose that an

For an undirected walk

Let

Since

Let

Let

Let

We consider two subcases: the subcase

First, we consider the subcase

Then, we consider the subcase

The proof for this subcase is similar to that of the Subcase

Let

Without loss of generality, we assume that

Without loss of generality, we assume that

First, we consider the subcase

Then, we consider the subcase

The proof for this subcase is similar to that of the Subcase

Let

The proof for this subcase is similar to that of Subcase

Without loss of generality, we assume that

Let

Since both

Let

Since both

The proof for this subcase is similar to that of the Subcase

Thus in each of the above subcases, there exists a pair of SSSD walks from

Since

Let

Let

Without loss of generality, we assume that

Since both

Let

Without loss of generality, we assume that

The proof for this subcase is similar to that of Subcase

The proof for this subcase is similar to that of Subcase

The proof for this subcase is similar to that of Subcase

From all the above subcases, there exists a pair of SSSD walks from

Illustrate for Subcase

Illustration for Subcases

Illustrate for Subcase

Let

Since

Let

By the Subcase 2.1 of Lemma 4.2 in [

By the Subcase

It remains to consider the following case.

Let

Without loss of generality, we assume that

Therefore,

Illustration for Case 3 of Lemma

Let

Assume that

Let

Since

If

Let

Let

Let

Let

Without loss of generality, we assume that

Let

Let

Therefore,

Let

Let

Without loss of generality, we assume that

Then, if

The verification for this subcase is similar to that of the Subcase 2.2 and is omitted.

Therefore,

Illustration for Lemma

Let

Let

Denote the vertex set of

Let

In this case,

In this case,

From all the above cases, there exists a pair of SSSD walks from

We get

Conversely, suppose that

This paper is supported by NSFC (61170311), Chinese Universities Specialized Research Fund for the Doctoral Program (20110185110020), and Sichuan Province Sci. & Tech. Research Project (12ZC1802).