1. Introduction
Let denote a digraph (directed graph) with n vertices, where the vertex set and the arc set Loops are permitted, but multiple arcs are not. A walk from x to y in we mean a sequence of vertices where each vertex in the sequence of vertices belongs to and a sequence of arcs where each arc in the sequence of arcs belongs to and the vertices and arcs are not necessarily distinct. The number of arcs in W is the length of the walk The notation means that there exists a walk of length k from x to The distance from vertex x to vertex y in D is written as (for short, ), which refers to the length of the shortest walk from x to If then a walk from x to y is a closed walk. A cycle is a closed walk from x to y with distinct vertices except for
Let
be any pair of vertices in a digraph
The digraph
D is called primitive, if there exists a positive integer
k such that there is a walk of length
k from
x to
This smallest such
k is denoted by
which is called the exponent of
The greatest common divisor of the lengths of all the cycles in
D is recorded as
It is well known (see [
1]) that
D is primitive if and only if
D is strongly connected and
Brualdi and Liu [
2] generalized the concept of exponent for a primitive digraph (primitive matrix). Let
D be a primitive digraph with
n vertices. The exponent of
D can be broken down into more local exponents [
3]. For any pair of vertices
let
denote the smallest integer
p such that there is a walk of length
t from
x to
z, for each integer
Since
D is a primitive digraph, then
is a finite number. For any vertex
the exponent of vertex
x is written as
which is the smallest integer
q so that for any vertex
there exists a walk of length
q from
x to
Moreover, for any vertex
and any integer
there is a walk of length
t from
x to
So, we have
Then, for any vertex
there is a walk of length
t from
x to
y for each integer
Therefore, we have
Let the vertices of
D be ordered as
such that
is called the
kth local exponent (generalized exponent) of
and it is denoted by
where
Then,
Furthermore, we have Obviously, the exponent of D equals That is, So, for a primitive digraph the local exponents of D are generalizations of the exponent of
Brualdi and Liu [
2] proposed a memoryless communication system. In the memoryless communication system represented by a primitive digraph
D of order
the
kth local exponent is the smallest time for each vertex to simultaneously hold all
k bits of the information. For more details, please refer to [
2,
3].
For any vertices
x and
y of a digraph
,
is an arc if and only if
is an arc, which is represented by
then such a digraph
D is called a symmetric digraph. An undirected graph (possibly with loops) can be viewed as a symmetric digraph. For some research on undirected graphs, please see [
4,
5,
6]. When
D is symmetric, the notation
indicates that there is a walk of length
k from
x to
Let be a symmetric digraph, we can regard D as an undirected graph. For convenience, undirected graph terms such as edges, edge set, etc., are used directly to describe a symmetric digraph. Then, let denote the set of undirected edges (edges) in Moreover, we assume that the notation represents that there is an edge in D with as end vertices.
Let be a symmetric digraph, where If for any vertices and , if and only if , then such a symmetric digraph D is called a doubly symmetric digraph. Moreover, and are called a pair of symmetrical edges, or is a symmetrical edge of where and The vertices , are called a pair of symmetric vertices, or is a symmetric vertex of where According to this definition, when n is odd, is symmetric to itself. If is a loop vertex, then and Therefore, for if is a loop, then is also a loop, the loops appear in pairs. A doubly symmetric digraph D is called a doubly symmetric primitive digraph provided D is primitive.
If a doubly symmetric primitive digraph D contains exactly d loops, then we call D a doubly symmetric primitive digraph with d loops. Let denote the set of all doubly symmetric primitive digraphs of order n. Let denote the set of all doubly symmetric primitive digraphs of order n with d loops, where d is an integer such that Obviously, we have
Let After deleting any pair of symmetrical edges and of the obtained digraph is not a doubly symmetric primitive digraph (that is, is not connected), then we call where Obviously, we have
For example, we consider the
kth local exponent of the graph
Let
,
Let
G is shown in
Figure 1.
We easily get , , , Then, we have Moreover, we have
Some studies [
7,
8,
9,
10,
11,
12] have investigated exponents and their generalization. Chen and Liu [
11] studied the
kth local exponent of doubly symmetric primitive matrices (primitive digraphs). Chen and Liu [
12] characterized the doubly symmetric primitive digraphs with the
kth local exponent reaching the maximum value. A doubly symmetric primitive digraph with
d loops is a special doubly symmetric primitive digraph. It is important to mention that the
kth local exponent of such a class of digraphs has not been studied before. Using graph theory methods, we obtain the upper bound of the
kth local exponent of digraphs in
where
Some studies have investigated the scrambling index [
13,
14,
15,
16] and generalized competition index [
17,
18,
19,
20,
21,
22,
23]. Several studies explored the generalized
-scrambling indices, please refer to [
24,
25,
26].
Let Let represent the set of d loop vertices in Let denote the set of d loops in Let , be any pair of vertices of the digraph If the walk from to in D is denoted as (for short, ), then is used to denote the length of the walk , and is used to denote the set of all vertices in this walk If there is a unique path from to in then let (for short, ) denote the unique path, and let (for short, ) denote the set of all vertices on the path. If then If a walk from to in D does not pass through a loop vertex, then let otherwise Similarly, if the unique path from to passes through a loop vertex, that is otherwise
For a vertex and a set let If let For any vertex and if let If T is a set, the notation is used to denote the number of all elements in The notation is used to denote the largest integer not greater than and the notation is used to denote the smallest integer not less than
In this paper, let and k be integers with , We give the upper bound of the kth local exponent of digraphs in where
2. The Upper Bound for the th Local Exponent of
In this section, let where
In the case of , we observe the exponent of any vertex in it is easy to get the following Proposition 1, let us omit the proof.
Proposition 1. Let and let be any vertex of then where
Lemma 1 (Lemma 3.3 [
2]).
Let D be a primitive digraph with n vertices. Then, where Remark 1. Lemma 1 is actually very useful. Next, we repeat the proof of Brualdi and Liu (see [2]). Since D is strongly connected, for any integer k such that there is a vertex x that is joined by an arc to one of the vertices with the k smallest exponents. Therefore, where Lemma 2. Let and let be any pair of vertices of Then,
Proof. For any vertex there is a walk of length t from to x, which is for each integer So, there is a walk of length s from to x, which is for each integer Therefore, □
Lemma 3. Let and let be any pair of vertices of If there exists a walk from x to y such that then
Proof. Let We consider the following.
Case 1 and
Suppose a walk from x to y through a loop vertex is denoted by where is a loop vertex and a is an integer such that Then, the length of the walk is The length of the walk is Similarly, we can easily conclude that there is a walk of length s from x to y, for each integer So,
Case 2 or
Similar to Case 1, it is easy to get that there is a walk of length s from x to y, for each integer So,
Therefore, the lemma holds. □
Lemma 4 (Lemma 1 [
23]).
Let If n is odd and x is any vertex of D, then Theorem 1. Let If n is odd and d is odd, then where
Proof. If d is odd, then is a loop vertex. Let x be any vertex. Then, a shortest path from x to goes through the loop vertex According to Lemma 4, we have Furthermore, according to Lemma 3, we have Further, we have So, Then, we have Further, according to Proposition 1, we have So, according to Lemma 1, we conclude Therefore, we have where □
Let and If D is a subgraph of such that and then where So, if we investigate the upper bound of the kth local exponent of digraphs in we only need to investigate the digraphs in
Referring to Definition 3 in [
23], we give the following Definition 1.
Definition 1. Let where n is odd, d is even such that There exist two connected subgraphs and of and satisfy and Where and Moreover,
Remark 2. Suppose n is odd and d is even that satisfies If then there are two connected subgraphs and of In addition, there is a unique path for any two different vertices in and respectively. Moreover, there is a unique path for any two different vertices in Let be any pair of vertices of D such that and then (see [23]). After removing d loops from the obtained graph is a tree. Therefore, D is a special tree with loops that satisfies if and only if , where Lemma 5 (Lemma 3 [
23]).
Let where n is odd, d is even such that Let be any pair of vertices of D such that and Then, there is a walk from x to y such that and Corollary 1. Let where n is odd, d is even and Let be any pair of vertices of D satisfying and Then,
Proof. According to Lemma 5, there is a walk from x to y passing through a loop vertex, and Moreover, according to Lemma 3, we have □
In Corollary 1, if and x isn’t a loop vertex, then we have
Theorem 2. Let If n is odd, d is even and then
- (1)
If then where
- (2)
Proof. We only need to consider Since d is even, then isn’t a loop vertex. Let x be any vertex. By Lemma 4, we have If the path from to x passes through a loop vertex, that is then If vertex x satisfies according to Corollary 1, we have Therefore, we have
- (1)
If then Then, we have According to Proposition 1, we have Then, according to Lemma 1, we can conclude that Therefore, we have where
- (2)
If then We have
Next, we construct a set such that and for any vertex holds. Suppose Then,
Suppose and If the vertex sequence of the unique path in from to is then the vertex sequence of the unique path in from to is Moreover, So, Therefore, we have Next, we prove that For any vertex if then If there is a vertex satisfying , then So, Moreover, and Then, we have Therefore, we have For any vertex next we consider
For any vertex let the walk from to x be If the walk passes through a loop vertex, we have If the walk doesn’t pass through a loop vertex, that is, Then, we have and So, if according to Corollary 1, we have Therefore, we have According to Proposition 1, we have Therefore, we have where
For any vertex such that , then There is a unique path for any pair of vertices in So for any vertex we have Suppose where We have So, we have Furthermore, according to Lemma 2, we can easily conclude that Since the conclusion is clearly established.
Therefore, the theorem holds. □
Referring to Definition 4 in [
23], we give the following Definition 2.
Definition 2. Let where n is even, d is even such that
- (1)
There exist two connected subgraphs and of and satisfy and Where and and Moreover,
- (2)
If Let and Suppose
Definition 3. Let where n is even, d is even such that In Definition 2(1), we give the following definition:
- (1)
Let be
then is a closed walk from to Let us write
- (2)
If let If let
Remark 3. Suppose n is even and d is even that satisfies If then there are two connected subgraphs and of Moreover, there is a unique path for any two different vertices in and respectively. Since D is connected, then there are edges and Then, If and are not loop vertices, then and If then , and is even such that Furthermore, if after removing d loops from the obtained graph is not a tree. According to Definitions 2 and 3, it is not difficult to see that and If then , and If then D is a special tree with loops that satisfies if and only if , where In fact, can be regarded as a special case of in
In Lemma 2 in [
23], let
where
n is even and
d be even such that
We can directly get the following Lemma 6.
Lemma 6. Let Let n be even and d be even such that Let x be any vertex of Then, for any vertex we have
Lemma 7 (Lemma 5 [
23]).
Let where n is even, d is even and Let be any pair of vertices of D such that If then there exists a walk from x to y such that and Corollary 2. Let where n is even, d is even such that Let be any pair of vertices of D satisfying If then
Proof. According to Lemma 7, there is a walk from x to y passing through a loop vertex, and Furthermore, according to Lemma 3, we have □
Theorem 3. Let If n is even, d is even and then
- (1)
If then where
- (2)
Proof. We only need to consider Let x be any vertex. Let us consider the following two cases.
Case 1
Since and then and
Case 1.1
Suppose Then, and are loop vertices. According to Lemma 6, we have Further, Then, Therefore, according to Proposition 1 and Lemma 1, we have where
- (1)
If then the conclusion is clearly established.
- (2)
If for then For we have
Case 1.2
Then,
For according to Corollary 2, we have
For any vertex then We have
For any vertex then Let the walk be Since then In addition, We have So Moreover, We have
Therefore, we have
- (1)
If then Then, we have Therefore, according to Proposition 1 and Lemma 1, where
- (2)
If then We have Next, we construct a set such that and for any vertex holds. Let Let Suppose Then, and Suppose and If the vertex sequence of the unique path in from to is then the vertex sequence of the unique path in from to is So, Therefore, we have Next, we prove that For any vertex if then If there is a vertex satisfying , we might as well assume Then So, Moreover, Then, Therefore, we have Let us assume Next, we consider
Case 1.2.1
Let be the walk from to which is If the walk passes through a loop vertex, we have If the walk does not pass through a loop vertex, we have and According to Corollary 2, then
Case 1.2.2
Let the walk be If the walk passes through a loop vertex, we have If the walk does not pass through a loop vertex. Then, Since then Moreover, we have So, we have Since and according to Corollary 2, then
Therefore, whether or we have So According to Proposition 1, we have Therefore, we have where For any vertex such that , then We have Furthermore, according to Lemma 2, we can conclude that Since the conclusion is clearly established.
Case 2
For it is equivalent to in , let us omit it. □
3. The th Local Exponents of and
In this section, we study the kth local exponents of the special graphs and .
Definition 4. Suppose Let and where are d loops arranged arbitrarily such that
Definition 5. Suppose d is even and . Let and let
From the definition of and we know that and Furthermore, there is a unique path for any two different vertices in and respectively.
Remark 4. In Figure 2, n can be either odd or even. In Figure 3, since n is odd and then is not a loop vertex. In Figure 4, since n is even and then and are not loop vertices. Theorem 4. If n is odd and d is odd, then where
Proof. Since d is odd, then is a loop vertex. We have Moreover, according to Lemma 2, we have where Further, where Therefore, the theorem now holds. □
Theorem 5. If n is odd, d is even and then
- (1)
If then where
- (2)
Proof. Since d is even, then is not a loop vertex.
- (1)
is shown in
Figure 2. Let
x be any vertex of
. If
then
If
according to Corollary 1, we have
Then,
If
then
Then, we have
Further,
Therefore,
Let
a be an integer satisfying
According to Lemma 2, we have
Further, we have
Then, by Proposition 1, where Therefore, we have where
- (2)
is shown in
Figure 3. If
then
Since
and
then
and
Then,
is not a loop vertex.
Suppose any integer a satisfies Since then Let x be any vertex of . Let the walk be If then If we have and
According to Corollary 1, we have Therefore, we have where Since and then Further, Therefore, where According to Proposition 1, if we have So, for we have Therefore, we have where
Suppose a satisfies We have Moreover, So, we can get where
Therefore, the theorem now holds. □
Theorem 6. If n is even, d is even and then
- (1)
If then where
- (2)
Proof. (1)
is shown in
Figure 2. We have
or
Let us assume
then
If
let us consider the following two cases.
Case 1
It is easy to see that Suppose a satisfies According to Lemma 2, we can get Further, According to Proposition 1, we have where Therefore, we have where
Case 2
Let x be any vertex of If then If according to Corollary 2, we have
Then, Further, we have Let a satisfy that Then, Further, we have
According to Proposition 1, where Therefore, we have where
(2)
is shown in
Figure 4. We have
or
Let us assume
then
If
then
Since
and
then
and
Then
and
are not loop vertices.
Suppose any integer a satisfies Since then Let x be any vertex of Let the walk be If then If we have and According to Corollary 2, we have So where Since and then Further, we have where According to Proposition 1, if we have So, for we have Therefore, we have where
Suppose a satisfies We have Moreover, So, we can get where
Therefore, the theorem now holds. □
It can be seen from Theorems 4–6, the upper bound for the kth local exponent of doubly symmetric primitive digraphs of order n with d loops can be reached, where .