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Some bounds of the generalized μ-scrambling indices of primitive digraphs with d loops
Journal of Inequalities and Applications volume 2021, Article number: 128 (2021)
Abstract
In 2010, Huang and Liu introduced a useful parameter called the generalized μ-scrambling indices of a primitive digraph. In this paper, we give some bounds for μ-scrambling indices of some primitive digraphs with d loops and the digraphs attained the sharp upper bounds are provided.
1 Introduction
For the research on the competition index, m-competition index, the scrambling index and the generalized μ-scrambling index, please refer to [1–3, 5, 6, 8, 9] and [7, 11], respectively. Cho et al. [6] defined the m-step competition graph of a digraph which is an extension of a competition graph. In 2009, Akelbek and Kirkland [2] defined and studied the scrambling index of a primitive digraph and provided an upper bound on the scrambling index of a primitive digraph. The m-competition index of a primitive digraph was introduced by Kim [8]. Kim investigated the m-competition index of a primitive digraph and gave an upper bound for the m-competition indices of primitive digraphs. In 2010, Huang and Liu [7] gave the definition of the generalized μ-scrambling indices for a primitive digraph which are a generalization of the scrambling index and m-competition index and they provided some bounds for the generalized μ-scrambling indices of some primitive digraphs. In this paper, we give some bounds for μ-scrambling indices of some primitive digraphs.
The outline of this paper is as follows: Some notation and notions used throughout this paper are introduced in Sect. 2. In Sect. 3, we study the generalized μ-scrambling indices of the primitive digraphs with d loops.
2 Definitions and terminology
In this section, we introduce some definitions,notations which are needed to use in the presentations and proofs of our main results in this paper.
A digraph D consists of a nonempty set \(V=V(D)\) and an arc set \(E=E(D)\). In D, loops are permitted but multiple arcs are not. A path \(P=x\rightarrow y\) is a sequence of edges \(\{(x, v_{1}), (v_{1}, v_{2}), \ldots , (v_{k-1}, y)\}\) in which all vertices are distinct. A cycle C is a closed path with the first and the last vertices coincided. A walk from x to y is a sequence of arcs: \(e_{1}, e_{2}, \dots , e_{k}\) such that the terminal vertex of \(e_{i}\) is the same as the initial vertex of \(e_{i+1}\) for \(i =1,2, \dots ,k-1\), denoted by \(W=x\rightarrow y\). The length of a walk or cycle is the number of arcs. A walk \(W=x\rightarrow y\) of length k is denoted by \(x\stackrel{{k }}{\rightarrow }y\). A cycle of length l is denoted by \(C_{l}\). The girth of D which has at least one cycle, is the length of a shortest cycle in D.
A digraph D is primitive with a walk of length k from each vertex x to each vertex y (not necessarily distinct). The digraph D is primitive if and only if D is strongly connected and the greatest common divisor of the lengths of its cycles is 1 (see [4]). For a positive integer s, the sth power of D, denoted by \(D^{(s)}\), is the digraph on the same vertex set \(V(D)\) and with an arc from i to j if and only if \(i\stackrel{{s }}{\rightarrow }j\) in D. The scrambling index \(k(D)\) of a primitive digraph D is the smallest positive integer k such that, for every pair of vertices u and v, there exists a vertex w such that \(u \stackrel{{k}}{\rightarrow }w\) and \(v\stackrel{{k}}{\rightarrow }w\) in D (see [2]).
Let D be a digraph with vertex set V and let k be a positive integer. A vertex w of D is a k-step common prey for u and v if \(u\stackrel{{k}}{\rightarrow }w\) and \(v\stackrel{{k}}{\rightarrow }w\). The k-step m-competition graph of D has the same vertex set of D and an edge between vertices u and v if and only if there are at least m distinct vertices \(v_{1},\ldots ,v_{m}\) in D such that \(u\stackrel{{k}}{\rightarrow }v_{i}\) and \(v\stackrel{{k}}{\rightarrow }v_{i}\) for \(i=1,2,\ldots ,m\) (see [6]). The m-competition index \(c(D,m)\) of a primitive digraph D is the smallest positive integer k such that, for every pair of vertices u and v, there are m distinct vertices \(v_{1},\ldots , v_{m}\) in D such that \(u\stackrel{{k}}{\rightarrow }v_{i}\) and \(v\stackrel{{k }}{\rightarrow }v_{i}\) for \(i=1,2,\ldots , m\) (see [2]). That is to say, the m-competition index of D is the smallest positive integer k such that the k-step m-competition graph is complete.
Let \(P_{n}\) denote the set of all primitive digraphs of order n.
Definition 2.1
([7])
Let \(D\in P_{n}\), and λ, μ be integers with \(1\leq \lambda ,\mu \leq n\). For \(X\subseteq V(D)\), let \(k^{(\mu )}_{X}\) be the smallest positive integer m such that there exist μ vertices \(w_{1},w_{2},\ldots ,w_{\mu }\) of D such that \(x\stackrel{{m }}{\rightarrow }w_{i}\) \((i=1,2,\ldots , \mu )\) in D for every vertex x of X. Then
are called the λth lower and upper μ-scrambling indices of D, respectively. For convenience, let \(k_{X}(D):=k^{(1)}_{X}(D)\), \(h(D,\lambda ):= h(D,\lambda ,1)\) and \(k(D,\lambda ) := k(D, \lambda , 1)\).
Since \(k(D,2)=k(D)\), in [7] Huang and Liu called \(h(D, \lambda ,\mu )\) and \(k(D, \lambda ,\mu )\) the generalized μ-scrambling indices, \(h(D,\lambda )\) and \(k(D,\lambda )\) the generalized scrambling indices of D in \(P_{n}\). As \(k(D,2,m)=c(D,m)\), the generalized μ-scrambling indices are also generalizations of the m-competition index.
3 Generalized μ-scrambling indices
In [7], Huang and Liu investigated generalized scrambling indices of the primitive digraphs with d loops. In this section, we study the generalized μ-scrambling indices of the primitive digraphs with d loops.
For a vertex subset \(X \subseteq V(D)\), define \(R_{t}^{D}(X)\) to be the set of vertices in D reachable from some vertices in X via a walk of length t.
Let d be an integer with \(1\leq d\leq n\) and let \(P_{n}(d)\) be the class of primitive digraphs with n vertices and d loops. Let \(L_{n,d}\) (\(1 \leq d\leq n\)) be the digraph with vertex set \(V(L_{n,d})=\{1,2,\ldots ,n \}\) and arc set
Theorem 3.1
Let \(D\in P_{n}(d)\) and \(1\leq \lambda ,\mu \leq n\).
and the bound can be attained by the digraph \(L_{n,d}\).
Proof
Since \(D\in P_{n}(d)\), there exists a loop vertex u such that there is a set Y of \(\lambda -1\) vertices whose distances to u are at most \(\lambda -1\). If \(\lambda +\mu < n+1\), let \(X=Y\cup \{u\}\). Then \(|X|=\lambda \). Since D is strongly connected and u is a loop vertex, the minimum number of vertices that can be reached from u at \((\mu -1)\)-step in D is μ. Therefore, \(|\bigcap _{x\in X}R^{D}_{\lambda +\mu -2}(\{x\})|\geq \mu \), which implies that \(h(D,\lambda ,\mu )\leq \lambda +\mu -2\).
If \(d\geq \lambda \) and \(\lambda +\mu \geq n+1\), let X be a vertex set which contains λ loop vertices. Since each vertex in X is a loop vertex, we have \(R_{n-1}^{D}(X)=V(D)\). Therefore, \(|R_{n-1}^{D}(X)|=|V(D)|=n\geq \mu \), which implies that \(h(D,\lambda ,\mu )\leq \lambda +\mu -2\).
If \(d<\lambda \), let Z be the vertex set of d loop vertices and \(X_{i}\subseteq (V(D)\setminus Z)\) be the vertex set of \(x_{i}\) vertices whose shortest distance to vertices of Z is i, where \(1\leq i\leq \lambda -d\). Assume \(\sum^{r}_{i=1}{x_{i}}\leq \lambda -d<\sum^{r+1}_{i=1}{x_{i}}\), where \(1\leq r\leq \lambda -d\). Let \(X=Z\cup X_{1}\cup \cdots \cup X_{r} \cup \bar{X}_{r+1}\), where \(\bar{X}_{r+1}\subseteq X_{r+1}\) contains \(\bar{x}_{r+1}\) vertices and \(\sum^{r}_{i=1}{x_{i}} +\bar{x}_{r+1}=\lambda -d \). Then \(|X|=\lambda \).
If \(d<\lambda \) and \(n+1 \leq \lambda +\mu \leq n+d\), since \(R_{n-1}^{D}(Z)=V(D)\) and \(R_{n-1}^{D}(X_{1}\cap \cdots \cap X_{r} \cap \bar{X}_{r+1})\) contains at least \(n-\sum^{r}_{i=1}{x_{i}} -\bar{x}_{r+1}=n-\lambda +d\) vertices, we have \(|R_{n-1}^{D}(X)|\geq n-\lambda +d\geq \mu \). Therefore, \(h(D,\lambda ,\mu )\leq \lambda +\mu -2\).
If \(d<\lambda \) and \(\lambda +\mu > n+d\), let \(k=\lambda +\mu -n-d\). Notice that \(R_{\lambda +\mu -d-1}^{D}(X_{i})=V(D)\), for \(1\leq i\leq k\). If \(k\geq r+1\), then \(|R_{\lambda +\mu -d-1}^{D}(X)|=|V(D)|=n\geq \mu \).
If \(k< r+1\), then \(R_{\lambda +\mu -d-1}^{D}(X)=\bigcap ^{r}_{i={k+1}}[R^{D}_{\lambda + \mu -d-1}(X_{i})]\cap R^{D}_{\lambda +\mu -d-1}(\bar{X}_{r+1})\). Since \(\bigcap ^{r}_{i={k+1}}[R^{D}_{\lambda +\mu -d-1}(X_{i})]\cap R^{D}_{ \lambda +\mu -d-1}(\bar{X}_{r+1})\) contains at least \(n-(\sum^{r}_{i=k+1}{x_{i}} +\bar{x}_{r+1})\) vertices, we have \(|R_{\lambda +\mu -d-1}^{D}(X)|\geq n-(\lambda -d-\sum^{k}_{i=1}{x_{i}}) \geq n-\lambda +d+k=n-\lambda +d+(\lambda +\mu -n-d)=\mu \). We thus arrive at \(h(D,\lambda ,\mu )\leq \lambda +\mu -2\).
On the other hand, consider the digraph \(L_{n,d}\). Lex X be a vertex set with λ vertices. If \(\lambda +\mu < n+1\), since \(R_{\lambda +\mu -3}^{L_{n,d}}(i)=\{i,\ldots ,n,\ldots ,\lambda +\mu +i-n-3 \}\), for \(n-d+1\leq i\leq n\) and \(R_{\lambda +\mu -3}^{L_{n,d}}(i)=\{n-d+1,\ldots ,n,\ldots ,\lambda + \mu +i-n-3\}\), for \(1\leq i\leq n-d+1\), we obtain \(|\bigcap _{x\in X}R^{L_{n,d}}_{\lambda +\mu -3}(\{x\})|\leq \mu -1\).
Noticing that \(R_{n-2}^{L_{n,d}}(i)=\{i,\ldots ,n,\ldots ,i-2\}\), for \(n-d+1\leq i\leq n\), and \(R_{n-2}^{L_{n,d}}(i)=\{n-d+1,\ldots ,n,\ldots ,i-2\}\), for \(1\leq i< n-d+1\). If \(d\geq \lambda \) and \(\lambda +\mu \geq n+1\), then, for any vertex \(x\in X\), \(R_{n-2}^{L_{n,d}}(\{x\})\) contains at most \(n-2\) vertices. Therefore, \(\bigcap _{x\in X}R^{L_{n,d}}_{n-2}(\{x\})\) contains at most \(n-\lambda -1\) vertices. As \(\lambda +\mu \geq n+1\), we have \(|\bigcap _{x\in X}R^{L_{n,d}}_{n-2}(\{x\})|\leq \mu -2\). Consequently, we obtain \(h(D,\lambda ,\mu )=\lambda +\mu -2\).
If \(d<\lambda \) and \(n+1 \leq \lambda +\mu \leq n+d\), we have for any set X, there is at least one vertex \(x\in X\) such that \(R_{n-2}^{L_{n,d}}(\{x\})\) contains at most \(n-3\) vertices. Thus, \(\bigcap _{x\in X}R^{L_{n,d}}_{n-2}(\{x\})\) contains at most \(n-\lambda -2\) vertices. Since \(\lambda +\mu \geq n+1\), we obtain \(|\bigcap _{x\in X}R^{L_{n,d}}_{n-2}(\{x\})|\leq \mu -3\). Therefore, \(h(D,\lambda ,\mu )=\lambda +\mu -2\).
If \(d<\lambda \) and \(\lambda +\mu > n+d\), we have \(R_{\lambda +\mu -d-2}^{L_{n,d}}(i)=V({L_{n,d}})\), for \(2n+2-\lambda -\mu \leq i\leq n\), and \(R_{\lambda +\mu -d-2}^{L_{n,d}}(i)=\{n-d+1,\ldots ,n,\ldots , \lambda +\mu +i-n-d-2\}\), for \(1\leq i<2n+2-\lambda -\mu \). Thus, for any vertex set X of λ vertices, there is a set \(Y\subseteq X\) of at least \(n+1-\mu \) vertices, such that, for any vertex \(y\in Y\), \(R_{\lambda +\mu -d-2}^{L_{n,d}}(\{y\})\) contains at most \(\lambda +\mu +y-n-2\) vertices, where \(1\leq y<2n+2-\lambda -\mu \). It follows that \(\bigcap _{y\in Y}R^{L_{n,d}}_{\lambda +\mu -d-2}(\{y\})\) containing at most \(\mu -1\) vertices. Therefore, \(|\bigcap _{x\in X}R^{L_{n,d}}_{\lambda +\mu -d-2}(\{x\})|\leq |\bigcap _{y \in Y}R^{L_{n,d}}_{\lambda +\mu -d-2}(\{y\})| \leq \mu -1\). It follows that \(h(D,\lambda ,\mu )=\lambda +\mu -2\). This completes the proof. □
Lemma 3.2
([10])
Let \(D\in P_{n}(d)\) and \(\emptyset \neq X\subseteq V(D)\). Then, for nonnegative integers i, j, t, k, we have \(R^{D}_{i}(X)=R^{D}_{i-j}(R^{D}_{j}(X))\) for \(i\geq j\), and \(|\bigcup ^{k}_{t=0}R^{D}_{t}(X)|\geq \min \{|X|+k,n\}\).
Theorem 3.3
Let \(D\in P_{n}(d)\) and \(1\leq \lambda ,\mu \leq n\). Then
and the bound can be attained by the digraph \(L_{n,d}\).
Proof
Let \(X\subseteq V(D)\) be a vertex set of any λ vertices. Set \(X=\{v_{1},v_{2},\ldots ,v_{\lambda }\}\).
Case 1. If \(\mu \leq d\).
For any vertex \(v_{i} \in X\), since \(D\in P_{n}(d)\subseteq P_{n}\), by Lemma 3.2,
where \(i=1,2,\ldots ,\lambda \). Let \(\frac{ d-\mu +1}{{\lambda }}=k^{\prime }+a\) where \(k^{\prime }\) is a nonnegative integer and \(0\leq a<1\). Therefore, if \(0< a<1\),
Since \(\lambda a\geq 1\),
If \(a=0\),
Since \(\lambda \geq 1\), we have
It follows that there are at least μ loop vertices \(u_{1},u_{2},\ldots ,u_{\mu }\) such that \(u_{i}\in \bigcap ^{\lambda }_{i=1}[\bigcup ^{n-k^{\prime }-1}_{t=0}R^{D}_{t}(\{v_{i} \})]\), where \(i=1,2,\ldots ,\mu \). That is to say,
Therefore, \(k(D,\lambda ,\mu ) \leq n-\lceil \frac{ d-\mu +1}{\lambda } \rceil \).
Consider the digraph \(L_{n,d}\). Let
Then we consider the following two subcases.
Subcase 1. If \(\lambda \leq d+1\).
When \(\lambda =1\), then \(t=n-d+\mu -1\). Noting that
then
When \(\lambda =2\), as
and
we have
When \(\lambda =3\), if \(n-2k^{*}-\mu +1< n-d+1\), then \(k^{*}=1\), \(t=n-1\). Let
where \(u\in V(L_{n,d})\setminus \{1, n-k^{*}-\mu +2\}\). Then \(|X|=3\). Since
and
we have
Thus,
If \(n-2k^{*}-\mu +1\geq n-d+1\), then \(n-2k^{*}-\mu +1< n-d+k^{*}+1< n-k^{*}-\mu +2\). Let
where \(u\in V(L_{n,d})\setminus \{1, n-k^{*}-\mu +2\}\). Then \(|X|=3\). Since
and
we have
which implies
When \(\lambda =4\), if \(n-2k^{*}-\mu +1< n-d+1\), then \(k^{*}=1\) and \(t=n-1\). Let
where \(u_{1},u_{2}\in V(L_{n,d})\setminus \{1,n-\mu +1\}\). Then \(|X|=4\). Since
and
we have
This implies
If \(n-d+k^{*}+1>n-2k^{*}-\mu +1\geq n-d+1\), letting
where \(w \in V(L_{n,d})\setminus \{1,n-k^{*}-\mu +2,n-d+k^{*}+1\}\), we have \(|X|=4\). Since
we have
Thus,
If \(n-d+k^{*}+1\leq n-2k^{*}-\mu +1\), letting
we have \(|X|=4\). Since
we have
When \(\lambda \geq 5\), if \(n-2k^{*}-\mu +1< n-d+1\), letting
where \(v_{i} \in V(L_{n,d})\setminus \{1,n-k^{*}-\mu +2\}\) for \(i=1,2,\ldots ,\lambda -2\), then \(|X|=\lambda \). Since
and
we have
which implies that
If \(n-d+k^{*}+1>n-2k^{*}-\mu +1\geq n-d+1\), letting
where \(v_{i}\in V(L_{n,d})\setminus \{1,n-d+k^{*}+1,n-k^{*}-\mu +2\}\), for \(i=1,2,\ldots ,\lambda -3\), then \(|X|=\lambda \). As
and
we have
Thus,
If \(n-(r+1)k^{*}-\mu +1< n-d+k^{*}+1\leq n-rk^{*}-\mu +1\) and \(2\leq r\leq \lambda -3\), letting
and
where \(v_{i} \in V(L_{n,d})\setminus \{1,n-d+k^{*}+1,n-k^{*}-\mu +2,n-2k^{*}- \mu +2,\ldots ,n-rk^{*}-\mu +2\}\) for \(i=1,2,\ldots ,\lambda -r-2\), then \(|X|=\lambda \). Since
and for \(1\leq i\leq r\),
we have
Thus,
If \(n-d+k^{*}+1\leq n-(\lambda -2)k^{*}-\mu +1\), letting
we have \(|X|=\lambda \). Since
for \(1\leq i\leq \lambda -2\)
and \(n-d+k^{*}+1>n-(\lambda -1)k^{*}-\mu +1\), we have
Therefore,
From the above, we have \(k(L_{n,d},\lambda ,\mu )\geq n-\lceil \frac{ d-\mu +1}{\lambda }\rceil \), it follows that \(k(L_{n,d},\lambda ,\mu )= n-\lceil \frac{ d-\mu +1}{\lambda } \rceil \).
Subcase 2. If \(\lambda > d+1\).
If \(\lambda > d+1\), then \(t=n-1\). It is easy to see that
for \(3\leq i\leq \lambda -d\),
and for \(i=n-d+1,\ldots ,n\),
Let
Then \(|X|=\lambda \). Since
and
we have \(\bigcap _{x\in X}R^{L_{n,d}}_{t-1}(\{x\})=\upphi \). Therefore,
it follows that
Case 2. If \(\mu > d+1\).
Let \(X \subseteq V(D)\) be a vertex set of any λ vertices. Set \(X=\{v_{1},v_{2},\ldots , v_{\lambda }\}\). For any vertex \(v_{i} \in X\), since \(D\in P_{n}(d)\subseteq P_{n}\), by Lemma 3.2,
where \(i=1,2,\ldots ,\lambda \). Therefore, each loop vertex \(u_{i}\in \bigcup ^{n-1}_{t=0}R^{D}_{t}(\{v_{i}\})\), where \(i=1,2,\ldots ,d\). Then
Since \(u_{1},u_{2},\ldots ,u_{d}\) are loop vertices, there are at least \(\mu -d\) vertices \(w_{1},w_{2},\ldots ,w_{\mu -d}\) and \(w_{i}\notin \{u_{1},u_{2},\ldots ,u_{d}\} \) such that
where \(i=1,2,\ldots ,\mu -d\). It follows that
We thus arrive at
Next we consider the digraph \(L_{n,d}\). Let \(X\subseteq L_{n,d}\) be a vertex set of λ vertices and set \(X=\{1,2,\ldots ,\lambda \}\). Let \(t=n+\mu -d-1\). Since for \(i=2,\ldots ,n-d\),
and for \(j=n-d+1,\ldots ,n\),
we have
which implies that
Therefore, \(k(L_{n,d},\lambda ,\mu )\geq n+\mu -d-1\). It follows that
Combining the proofs of Cases 1 and 2, the theorem follows as expected. □
Theorem 3.4
Let \(D\in P_{n}\) with girth s. Then
Proof
Let \(C_{s}\) be a directed cycle of length s in \(D^{(s)}\). Consider the digraph \(D^{(s)}\). Choose any r vertices \(w_{1},w_{2},\ldots ,w_{r}\) of \(C_{s}\). Let \(\frac{n-\mu }{r}=k+\frac{{b}}{r}\) where \(0\leq b< r-1\). In \(D^{(s)}\), since \(w_{i}\) is a loop vertex, \(|R^{D^{s}}_{n-k-1}\{w_{i}\}|\geq n-k=n-\frac{n-\mu }{r}+\frac{{b}}{r}\) where \(i=1,2,\ldots ,r\). Therefore
It follows that
For any λ vertices \(v_{1},v_{2},\ldots ,v_{\lambda }\in V(D)\), there is a walk of length \(n-s\) from \(v_{i}\) to a vertex \(u_{i}\) of \(C_{s}\) where \(i=1,2,\ldots ,\lambda \). If \(\lambda \leq s\), then \(|u_{1},u_{2},\ldots ,u_{\lambda }|\leq \lambda \) and if \(\lambda > s\), then \(|u_{1},u_{2},\ldots ,u_{\lambda }|\leq s\). Hence,
□
4 Conclusions
In this paper, we studied μ-scrambling indices of primitive digraphs and gave some bounds for the λth lower and upper μ-scrambling indices of primitive digraphs with d loops. However, the digraphs attaining the sharp upper bounds are not determined completely. For a general given primitive digraph, its μ-scrambling indices are not given. It would be nice to settle these problems in further research.
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Acknowledgements
The first author wishes to thank Professor Jun He for his guidance.
Funding
The work was supported by the Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant No. KJ1600512), National Natural Science Foundation of China (Grant No. 11701058) and Team Building Project for Graduate Tutors in Chongqing (Grant No. JDDSTD201802).
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Zhang, L., Mou, GF., Liu, F. et al. Some bounds of the generalized μ-scrambling indices of primitive digraphs with d loops. J Inequal Appl 2021, 128 (2021). https://doi.org/10.1186/s13660-021-02667-y
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DOI: https://doi.org/10.1186/s13660-021-02667-y