Is there a name for the matrix $D+A$ in graph theory?

Question

In graph theory, we have the definition of Laplacian as $L=D-A$, where D is the degree matrix and A is the adjacency matrix.

But is there a name for the matrix $L'=D+A$?

Obviously, $L = |L'|$ and having all the elements to be positive.

Answer 1

The matrix $D+A$ is called the signless Laplacian matrix. (See, e.g., section 1.1 "Matrices associated with a graph" of Spectra of graphs by Brouwer and Haemers.)

Notably, just as $L=D-A$ has $u^{\mathsf T} L u = \sum_{xy \in E} (u_x - u_y)^2$, $Q = D + A$ has $u^{\mathsf T}Q u = \sum_{xy \in E} (u_x + u_y)^2$, and we can write $Q$ as $MM^{\mathsf T}$ where $M$ is the incidence matrix of the graph.

The signless Laplace matrix is useful for dealing with bipartite graphs and subgraphs. For example:

• (Proposition 1.3.9 in the source above) the multiplicity of $0$ as an eigenvalue of $D+A$ is the number of bipartite components of the graph;
• (Proposition 1.3.10) a graph is bipartite if and only if $D+A$ and $D-A$ have the same eigenvalues.