Bounds for the signless Laplacian energy of digraphs
The paper under review gives upper and lower bounds for the signless Laplacian energy of finite directed graphs without loops and multiple arcs but perhaps with a pair of oppositely directed arcs joining the same pair of vertices. The signless Laplacian is defined to be \(Q=D+A,\) where \(D\) is the diagonal matrix with outdegrees of vertices along the diagonal and \(A\) is the adjacency matrix. The energy of the signless Laplacian is defined to be the sum of \(\left| q_{i}-\frac{a}{n}\right| ,\) where \(q_{i}\) is the \(i\)th eigenvalue of the signless Laplacian, while \(a\) is the number of arcs and \(n\) the number of vertices. For example, Theorem 2.4 says that, if \(d_{i}^{+}\) denotes the outdegree of the \(i\)th vertex, the energy of the signless Laplacian is less than or equal to \(\;\sqrt{M_{1}n},\) where \(M_{1}=a+\sum \limits_{i=1}^{n}\sum\limits_{i=1}^{n} \left( d_{i}^{+}-\frac{a}{n}\right) ^{2}.\) Theorem 2.7 says that if \(c_{2}\) is the number of directed closed walks of length \(2,\) the energy of the signless Laplacian is greater than or equal to \(\sqrt{2M},\) where \(M=c_{2}+M_{1}-a.\)
scientific article; zbMATH DE number 6843790
scientific article