Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A mixed graph is obtained from an undirected graph by orienting a subset of its edges. The Hermitian adjacency matrix of a mixed graph  M  of order  n  is an  n  ×  n    matrix   H(M)=(h kl ),      H   (  M  )   =   (   h   k  l    )   ,         where   h kl =−h lk =i       h   k  l    =  −   h   l  k    =  i         (            i  =    −  1          ) if there exists an orientation from  v k   to  v l     and   h kl =h lk =1       h   k  l    =   h   l  k    =  1         if there exists an edge beten  v k   and  v l     but not exist any orientation, and   h kl =0       h   k  l    =  0         otherwise. Let   D(M)=diag(d 1 ,d 2 ,…,d n )      D   (  M  )   =  d  i  a  g   (   d  1   ,   d  2   ,  …  ,   d  n   )          be a diagonal matrix where  d i   is the degree of vertex  v i   in the underlying graph  M u    . Hermitian matrices   L(M)=D(M)−H(M),      L  (  M  )  =  D  (  M  )  −  H  (  M  )  ,          Q(M)=D(M)+H(M)      Q  (  M  )  =  D  (  M  )  +  H  (  M  )         are said as the Hermitian Laplacian matrix, Hermitian quasi-Laplacian matrix of mixed graph  M , respectively. , it is shown that they are positive semi-definite. Moreover,  characterize the singularity of them. In addition, an expression of the determinant of the Hermitian (quasi-)Laplacian matrix is obtained.
['Guihai Yu', 'Xin Liu', 'Hui Qu']
I am writing a research paper on a idea Singularity of Hermitian (quasi-)Laplacian matrix of mixed graphs
[{"from": "human", "value": "I am writing a research paper on a idea Singularity of Hermitian (quasi-)Laplacian matrix of mixed graphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A mixed graph is obtained from an undirected graph by orienting a subset of its edges. The Hermitian adjacency matrix of a mixed graph  M  of order  n  is an  n  \u00d7  n \u00a0  matrix   H(M)=(h kl ),      H   (  M  )   =   (   h   k  l    )   ,         where   h kl =\u2212h lk =i       h   k  l    =  \u2212   h   l  k    =  i         (            i  =    \u2212  1          ) if there exists an orientation from  v k   to  v l  \u00a0  and   h kl =h lk =1       h   k  l    =   h   l  k    =  1         if there exists an edge beten  v k   and  v l  \u00a0  but not exist any orientation, and   h kl =0       h   k  l    =  0         otherwise. Let   D(M)=diag(d 1 ,d 2 ,\u2026,d n )      D   (  M  )   =  d  i  a  g   (   d  1   ,   d  2   ,  \u2026  ,   d  n   )          be a diagonal matrix where  d i   is the degree of vertex  v i   in the underlying graph  M u  \u00a0 . Hermitian matrices   L(M)=D(M)\u2212H(M),      L  (  M  )  =  D  (  M  )  \u2212  H  (  M  )  ,          Q(M)=D(M)+H(M)      Q  (  M  )  =  D  (  M  )  +  H  (  M  )         are said as the Hermitian Laplacian matrix, Hermitian quasi-Laplacian matrix of mixed graph  M , respectively. , it is shown that they are positive semi-definite. Moreover,  characterize the singularity of them. In addition, an expression of the determinant of the Hermitian (quasi-)Laplacian matrix is obtained."}]