The spectral radius and maximum outdegree of irregular digraphs☆
Introduction
Let be a digraph with vertex set and arc set . The order of is and the size of is . If there is an arc from to , we indicate this by writing , call (resp. ) the head (resp. the tail) of . An arc with identical head and tail is called a loop. A digraph is simple if it has no loops and no two of its arcs have the same head and tail. A directed path in the digraph is a finite non-null sequence , whose terms are alternately vertices and arcs, such that, for , the arc has head and tail , and the vertices are distinct. A directed path is often represented simply by its vertices . A digraph is strongly connected if for every pair of vertices , there exists a directed path from to . Let denote the complete digraph on vertices in which two arbitrary distinct vertices , there are arcs and . Throughout this paper, we only consider the simple digraphs which have the maximum vertex indegree is no more than the maximum vertex outdegree.
Let be a simple strongly connected digraph. For any vertex , and are called the sets of out-neighbors and in-neighbors of , respectively. Let denote the outdegree of the vertex and denote the indegree of the vertex in the digraph . The maximum vertex outdegree is denoted by , or simply , the minimum vertex outdegree by , or simply if it is clear from the context. If , then is regular. The distance from to in , denoted by , is defined to the length of the shortest directed path from to . The diameter of is the maximum over all ordered pair of , .
The vertex connectivity of a strongly connected digraph can be defined in two ways. The vertex connectivity from vertex to can be defined either as the maximum number of pairwise internally vertex disjoint directed paths from to or as the minimum number of vertices which when removed from the digraph render inaccessible from along a directed path. For a strongly connected digraph , the vertex connectivity of , denoted by , is the minimum over all ordered pair of vertices . It is also therefore the smallest numbers of vertices whose deletion yields the resulting digraph non-strongly connected. For an integer , is called -connected if . The arc connectivity of , denoted by , is the smallest numbers of arcs whose deletion yields the resulting digraph non-strongly connected. For an integer , is called -arc-connected if .
Let denote the adjacency matrix of a digraph , where whenever , otherwise. Let be the diagonal matrix with outdegrees of the vertices of . In this paper, we study the convex linear combinations of and , defined as It is easy to see that, Since is essentially equivalent to , in this paper we take as an exact substitute for . The spectral radius and signless Laplacian spectral radius of digraphs have been well studied in the literature, see [1], [2], [3], [4], [5]. The spectral radius of , i.e., the largest modulus of the eigenvalues of , is called the spectral radius of , denoted by . If and is a strongly connected digraph, then is nonnegative irreducible, and by Perron Frobenius Theorem [6], is an eigenvalue of , and there is a positive unit eigenvector corresponding to . The positive unit eigenvector corresponding to is called the Perron vector of . Unless stated otherwise, we assume that in the rest of this paper.
The spectral radius of graphs has been studied in the literature, see [7], [8], [9], [10], [11], [12], [13], [14], [15], but there is not much known about digraphs. Recently, Liu et al. [16] determined the unique digraph which attains the maximum (resp. minimum) spectral radius among all strongly connected bicyclic digraphs, they also characterized the digraph which has the maximal spectral radius among all strongly connected digraphs with given dichromatic number. Xi et al. [17] determined the digraphs which attain the maximum (or minimum) spectral radius among all strongly connected digraphs with given parameters such as girth, clique number, vertex connectivity and arc connectivity.
It is well known that the adjacency spectral radius, denoted by , of a connected regular graph is the maximum degree . has been considered as a measure of irregularity for a graph [18]. Some estimates on for a connected irregular graph have been obtained in many papers. We will give some known results on of a connected irregular graph . Let be a connected irregular graph with vertices, edges, maximum degree and diameter . In [19], Cioabǎ et al. showed that In [20], Cioabǎ proved that In [21], Chen and Hou proved that if is a -connected irregular graph with vertices, edges and maximum degree , then
Let be a connected irregular graph with vertices and diameter , be the signless Laplacian spectral radius of the graph . In [22], Ning et al. showed that
In [23], Shiu et al. proved that if is a -connected irregular graph with vertices, edges and maximum degree , then
In [24], Ning et al. proved that if is a -connected irregular graph with vertices, edges, maximum degree and minimum degree , then
However, there is no much known about digraphs. Motivated by these known estimates on and and some methods used in estimating them, we consider the following question:
How small can be when is a strongly connected irregular digraph?
Section snippets
Main results
Lemma 2.1 If , then with equality if and only if .[25]
Theorem 2.2 Let be a strongly connected irregular digraph of order , size , diameter , maximum outdegree , minimum outdegree and average outdegree . Then
Proof Let be the Perron vector of corresponding to . Clearly, . Suppose that , are two vertices satisfying and , respectively. Let be the shortest directed path
Acknowledgments
The authors are very grateful to the anonymous referees for their valuable comments and suggestions, which lead to a great improvement in the presentation of the manuscript.
References (25)
- et al.
Spectral radius and signless Laplacian spectral radius of strongly connected digraphs
Linear Algebra Appl.
(2014) - et al.
On the spectral radius of simple digraphs with prescribed number of arcs
Discrete Math.
(2015) - et al.
The maximum Perron roots of digraphs with some given parameters
Discrete Math.
(2013) - et al.
The (distance) signless Laplacian spectral radius of digraphs with given arc connectivity
Linear Algebra Appl.
(2019) - et al.
On the signless Laplacian spectral radius of weighted digraphs
Discrete Optim.
(2019) - et al.
The -spectral radius of trees and unicyclic graphs with given degree sequence
Appl. Math. Comput.
(2019) - et al.
Graphs determined by their -spectra
Discrete Math.
(2019) - et al.
A note on the -spectral radius of graphs
Linear Algebra Appl.
(2018) - et al.
On the -characteristic polynomial of a graph
Linear Algebra Appl.
(2018) - et al.
On the -index of graphs with pendent paths
Linear Algebra Appl.
(2018)
A note on the positive semidefiniteness of
Linear Algebra Appl.
On the -spectral radius of a graph
Linear Algebra Appl.
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Supported by the National Natural Science Foundation of China (No. 11871398), the Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2018JM1032) and the Fundamental Research Funds for the Central Universities, PR China (No. 3102019ghjd003).