1. Introduction
Let be a digraph with vertex set and arc set . If there is an arc from the vertex to vertex , then is said to be adjacent to , and we denote this arc by writing . For the arc , the first vertex is its tail, and the second vertex is its head. For any vertex , the outdegree of is the number of arcs of which is the tail. A digraph G is simple if it has no loops and multiple arcs. A digraph G is strongly connected if for every pair of vertices , there is a directed path from to and one from to . Throughout this paper, we consider finite, simple strongly connected digraphs.
A digraph is a r-partite digraph if its vertices can be partitioned into r arcless sets. Let denote the complete r-partite digraph of order n, whose partition sets are of size or . Let denote the complete digraph of order n in which for two arbitrary distinct vertices , there are arcs and in .
For a strongly connected digraph G, the distance from the vertex to vertex , denoted by , is defined as the length of the shortest directed path from to in G. The diameter of the strongly connected digraph G, denoted by , is the maximum over all ordered pairs of vertices .
Let be the distance matrix of G, where is the distance from to . For vertex , the transmission of in G, denoted by or , is defined as the sum of distances from to all other vertices in G, that is, . Let be the diagonal matrix with vertex transmissions of G in the diagonal and zeroes elsewhere. Then, the distance signless Laplacian matrix of G is the matrix . For any real number , the general distance matrix of G is defined as the matrix .
Spectral graph theory is a fast-growing branch of algebraic graph theory. One of the central issues in spectral graph theory is as follows: For a graph matrix, determine the maximization or minimization of spectral invariants over various families of graphs. Recently, the spectral radius of the distance matrix, the related distance signless Laplacian matrix, and the general distance matrix of digraphs have received increasing attention, see [
1,
2,
3,
4,
5,
6]. In particular, Lin et al. [
6] characterized the extremal digraphs with minimum distance spectral radius among all digraphs with given vertex connectivity. Xi and Wang [
4] determined the strongly connected digraphs minimizing distance spectral radius among all strongly connected digraphs with a given diameter
d, for
. Li et al. [
2] characterized the digraph minimizing the distance signless Laplacian spectral radius among all strongly connected digraphs with given vertex connectivity. Xi et al. [
7] characterized the extremal digraph achieving the minimum distance signless Laplacian spectral radius among all strongly connected digraphs with given arc connectivity. Xi et al. [
8] proposed to study the generalized distance spectral radius of strongly connected digraphs, and they determined the digraphs which attain the minimum
spectral radius among all strongly connected digraphs with given parameters such as dichromatic number, vertex connectivity, or arc connectivity.
The matrices related to simple, undirected graphs are symmetric matrices, which have well-defined spectral properties. However, unless the digraph
G is symmetric, it is not the case that
for all vertices
of
G; that is, the symmetric property does not hold for directed distance. In order to produce a symmetric matrix from a digraph, Chartrand and Tian [
9] proposed one metric on strongly connected digraphs: sum distance defined as
; then, one can obtain a symmetric matrix on a strongly connected digraph, which is called the sum distance matrix.
Let G be a strongly connected digraph of order n, be the vertex set, and be the arc set. The sum distance matrix of G is the matrix , where . For vertex , the sum transmission of in G, denoted by or , is the row sum of the sum distance matrix corresponding to vertex , which is . A strongly connected digraph G is r-sum transmission regular if for each ; otherwise, G is not sum transmission regular. Let be the diagonal matrix with vertex sum transmissions of G in the diagonal and zeroes elsewhere. The matrix is called the sum distance Laplacian matrix of G, and the matrix is called the sum distance signless Laplacian matrix of G. Note that all the matrices , and are symmetric. Therefore, they have real eigenvalues. The eigenvalues of are called the sum distance eigenvalues of G, denoted by . The eigenvalues of are called the sum distance Laplacian eigenvalues of G, denoted by . The eigenvalues of are called the sum distance signless Laplacian eigenvalues of G, denoted by .
For any real number
, similar to [
8,
10], Xu and Zhou [
11] proposed to study the general sum distance matrix of
G:
Obviously,
,
,
, and
. The matrix
enables a unified study of
and
. The eigenvalues of
are called the general sum distance eigenvalues of
G, denoted by
. The spectral radius of
, i.e., the largest eigenvalue of
, is called the general sum distance spectral radius of
G, denoted by
. For a strongly connected digraph
G,
is a non-negative irreducible matrix. Based on the Perron Frobenius Theorem [
12],
is a simple eigenvalue of
, and there is a positive unit eigenvector corresponding to
. The positive unit eigenvector corresponding to
is called the Perron vector of
.
In this paper, we first give some spectral properties of . We also characterize the digraph minimizes the general sum distance spectral radius among all strongly connected r-partite digraphs. Moreover, for digraphs that are not sum transmission regular, we give a lower bound on the difference between the maximum vertex sum transmission and the general sum distance spectral radius.
2. Preliminaries and Basic Properties of
For
,
denotes the
i-th largest eigenvalue of Hermitian matrix
A. In the following, we give Weyl’s inequalities [
12] for eigenvalues of Hermitian matrices, and the equality case was first established by So in [
13].
Lemma 1 ([
12] Theorem WS).
Let A and B be two Hermitian matrices of order n, and also let . Then,- (1)
- (2)
Moreover, either of the equality holds if and only if there exists a unit vector that is an eigenvector to each of the three eigenvalues involved.
Lemma 2 ([
14] Interlacing Theorem).
Let A be a real symmetric matrix of order n, and B be a principal submatrix of A with order s. Then, Let
M be a real matrix of order
n described in the following block form:
where the diagonal blocks
are
matrices for any
and
. For any
, let
denote the average row sum of
, i.e.,
is the sum of all entries in
divided by the number of rows. Then,
is called the quotient matrix of
M. In addition, if for each pair
,
has a constant row sum, then
is called the equitable quotient matrix of
M.
Lemma 3 ([
15]).
Let be defined as above, be the equitable quotient matrix of M. Then, the spectrum of B is contained in the spectrum of A. Moreover, if M is a non-negative matrix, then , where is the spectral radius of B. Proposition 1. Let , G be a strongly connected digraph of order n with and . Then, for any , Proof. Notice that
, and
. Set
and
, applying the (1) of Lemma 1, one obtains
Hence, the inequality holds. □
Proposition 2. Let G be a strongly connected digraph of order with , where . Then, for any , Proof. Taking
. Then, the diagonal
-entry of
M for
is
, and the non-diagonal
entry of
M is
. Moreover, for
, it is easy to know that
M is a diagonally dominant matrix. Therefore, we have
M as a diagonally dominant with non-negative diagonal entries when
, so it is a positive semi-definite matrix, which implies that the least eigenvalue is at least zero. Hence, for
, by Lemma 1, we obtain
Thus, we achieve the desired result. □
For the complete digraph , we have , where is the identity matrix and is the matrix in which every entry is 1. Thus, , where denotes the spectrum of the matrix , and denotes is an eigenvalue of multiplicity .
Proposition 3. Let G be a strongly connected digraph of order . Then, for any ,with equality if and only if . Proof. By Proposition 2, we obtain
. Suppose that
and
. Then
G is (isomorphic to) a subdigraph of
for some
. Without loss of generality, we assume that
. Let
,
. Then the equitable quotient matrix of
corresponding to the partition
is
According to Lemma 3,
is an eigenvalue of
for
. Let
. Then,
Hence,
and
. Since
is strictly decreasing when
and strictly increasing when
,
and
. Furthermore,
. Based on Proposition 2,
, a contradiction. Thus, for any
,
with equality if and only if
. □
For a strongly connected digraph
G of order
n,
is the general sum distance matrix of
G,
is the sum transmission of
in
G. Let
be a real column vector; then,
3. The General Sum Distance Spectral Radius of Strongly Connected Digraphs
In this section, we study the general sum distance spectral radius of strongly connected digraphs.
Lemma 4 ([
12]).
Let be an non-negative matrix with spectral radius , and let be the i-th row sum of P, i.e., . Then,Moreover, if P is irreducible, then any equality holds if and only if .
From Lemma 4, we have the following theorem:
Theorem 1. Let G be a strongly connected digraph with . Then,where . Moreover, if , then either one equality holds if and only if G is sum transmission regular. Proof. Since
is the diagonal matrix with vertex sum transmissions of
G in the diagonal and zeroes elsewhere, with a simple calculation, we obtain the
i-th row sum of
as
Take . Using Lemma 4, the required result follows.
For
, suppose that either of the equalities holds; then, Lemma 4 implies that the row sums of
are all equal. That is, for any vertices
,
Let
and
denote the maximum and minimum vertex sum transmissions of
G, respectively. Without a loss of generality, assume that
and
. One can easily see that
and
. Thus, we obtain
which implies that
for
. Therefore,
G is the sum transmission regular.
Conversely, if
G is an
r-sum-transmission-regular digraph, then
. On the other hand, through a simple calculation, we obtain
for any
. Therefore, both equalities hold. □
Lemma 5. Let G be a strongly connected digraph with , where . Then, for any , Recall that denotes the complete r-partite digraph of order n, whose partition sets are of size or . Note that . It is known that has maximum number of arcs among all r-partite digraphs of order n. Then, we have , which has the minimum among all strongly connected r-partite digraphs of order n. We will consider the case first.
Theorem 2. Let G be a strongly connected bipartite digraph with order n. Then, for any ,with equality if and only if . Proof. Suppose that
G is a strongly connected bipartite digraph of order
n with minimum
among all strongly connected bipartite digraphs of order
n; then, based on Lemma 5,
G is a complete bipartite digraph. Let
and
be the partitions of the vertex set of
G, where
,
,
and
. Let
be the Perron vector of
. One can easily infer that the entries of
corresponding to vertices in the same partition set have the same value, say,
for
. Thus,
can be written as
. From
, and we have
Combining the above two equations, we have
Since for , is decreasing for and increasing for . Then is minimum whenever , that is . Therefore, we have , with equality if and only if . □
In general, for , we have the following theorem:
Theorem 3. Let G be a strongly connected r-partite digraph of order n, where . Then, for any ,with equality if and only if . Proof. Suppose that
G is a strongly connected
r-partite digraph of order
n with minimum
among all strongly connected
r-partite digraphs of order
n; then, based on Lemma 5, we have
G, which is a complete
r-partite digraph. Let
be the partition sets of
, where
and
. Let
be the Perron vector of
. One can easily infer that the entries of
corresponding to vertices in the same partition set have the same value, say,
for
. From
, for any
, we obtain
From the above equation, we have
. Let
. Then, we obtain
, which implies
Let
one can easily see that
for
. That is
is increasing for
. Then, we obtain that
is attained if and only if
or
for any
. In fact, the minimum of (
5) can be attained, as there are finitely many vectors
satisfying the constraints. Suppose that the minimum of (
5) is attained for some
, and through symmetry, we can assume that
. If
, the calculation is completed. In the following, we assume
for a contradiction. Taking
According to the mean value theorem, there exist
and
such that
Since
is increasing for
and
,
. Therefore,
that is
which is contrary to the assumption that
is minimum. Therefore,
or
for any
.
Let
and
be the sizes of the partition sets of
, that is
or
for any
and
. Using Equation (
4), we obtain
Thus, , with equality if and only if or for any . □