Sentence examples for matrix of degrees from inspiring English sources

Let (G= V, E)) be a simple undirected graph with vertex set (V=V(G)= {v_{1}, v_{2}, ldots, v_{n}}) and edge set (E=E(G)), where n is called the order of G. Let (A(G)) be the adjacency matrix of a graph G and let (D(G =operatorname {diag}(d_{G}(v_{1}), d_{G}(v_{2}),ldots, d_{G}(v_{n}))) be the diagonal matrix of degrees of G, where (d_{G} v)) or simply (d v)) denotes the degree of a vertex v in G.

The final formulas contain only Bézier points and matrix of degree elevation.

A differentiation matrix, (mathbf {D}), of size (n+1) in a certain basis is a nilpotent matrix of degree (n+1).

In the following, we prove that widehat{A}:=left [ textstylebegin{array}c@{quad}c@ A_{1}& A_{2} A_{3} & A_{4 end{array}displaystylend{array}displaystylent matrightf degree τ.

and thus P S : = | S 〉 〈 S | ∈ ( H ν ⊗ 1 + 1 ⊗ H ν ) ′ holds jointly for all ν ∈ { 0 ; 1, …, m } ; 0 N denotes the zero matrix of degree N. In this section, we present a straightforward criterion for pure-state controllability of quasifree fermionic systems with d modes.

Let (D(G)) be the diagonal matrix of vertex degrees of G.

Let (D(M_{U})) be the diagonal matrix of vertex degrees of (M_{U}).

Let (A(G)) and (D(G)) be the adjacency matrix and the diagonal matrix of vertex degrees of G, respectively.

If M has no isolated vertices, then (R_{H}(M =D(M_{U})^{-frac {1}{2}}H(M D(M_{U})^{-frac{1}{2}}) is applicable, where (D(M_{U})) is the diagonal matrix of vertex degrees.

From a probabilistic point of view, the more natural matrix to associate with a graph is the Laplacian matrix L G), which is the infinitesimal generator of the natural random walk on the graph and is given by A(G) - D(G), where D G) is the diagonal matrix of vertex degrees.

In order to prove results for the adjacency and Laplacian matrices simultaneously, we define for a tree T and arbitrary real numbers y and z the generalized Laplacian L ˜ (T ) : = y D (T ) + z A (T ) (recall that A(T) is the adjacency matrix and D(T) is the diagonal matrix of vertex degrees).