On the Skew Laplacian Energy of a Digraph 1 INTRODUCTION

On the skew Laplacian energy of a digraph 1911
Then
ESL(G) =
m
ESL(Gi).
i=1
Theorem 2.6. Let G be a simple digraph with vertex degrees are d1,d2, ..., dn.
Then we have
ESL(G) =
n
di(di − 1). (2.1)
i=1
Proof. Let G be a simple digraph with vertex set V (G) ={v1,v2, ..., vn} and
d(vi) =di for i =1, 2, ..., n. Let λ1,λ2,λ3, ..., λn be the eigenvalues of the
Laplacian matrix L(G) =D(G) − S(G) where D(G) =diag(d1,d2, ..., dn) and
n
S(G) is the skew-adjacency matrix of digraph G. We have λi = sum of
determinants of all 1 × 1 principal submatrices of L(G) = trace of L(G)
n
= di. Note that
λiλj = sum of determinants of all 2 × 2 principal
i=1
submatrices of L(G)
=
det
i