On the General Sum Distance Spectra of Digraphs

Let G be a strongly connected digraph, and d G ( v i , v j ) denote the distance from the vertex v i to vertex v j and be defined as the length of the shortest directed path from v i to v j in G . The sum distance between vertices v i and v j in G is defined as s d G ( v i , v j ) = d G ( v i , v j ) + d G ( v j , v i ) . The sum distance matrix of G is the n × n matrix S D ( G ) = ( s d G ( v i , v j ) ) v i , v j ∈ V ( G ) . For vertex v i ∈ V ( G ) , the sum transmission of v i in G , denoted by S T G ( v i ) or S T i , is the row sum of the sum distance matrix S D ( G ) corresponding to vertex v i . Let S T ( G ) = diag ( S T 1 , S T 2 , … , S T n ) be the diagonal matrix with the vertex sum transmissions of G in the diagonal and zeroes elsewhere. For any real number 0 ≤ α ≤ 1 , the general sum distance matrix of G is defined as S D α ( G ) = α S T ( G ) + ( 1 − α ) S D ( G ) . The eigenvalues of S D α ( G ) are called the general sum distance eigenvalues of G , the spectral radius of S D α ( G ) , i.e., the largest eigenvalue of S D α ( G ) , is called the general sum distance spectral radius of G , denoted by μ α ( G ) . In this paper, we first give some spectral properties of S D α ( G ) . 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.