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A vertex with a large degree, also called a heavy node, results in a large diagonal entry in the Laplacian matrix dominating the matrix properties. Normalization is aimed to make the influence of such vertices more equal to that of other vertices, by dividing the entries of the Laplacian matrix by the vertex degrees.
In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid.For the case of a finite-dimensional graph (having a finite number of edges and vertices), the discrete Laplace operator is more commonly called the Laplacian matrix.
The algebraic connectivity (also known as Fiedler value or Fiedler eigenvalue after Miroslav Fiedler) of a graph G is the second-smallest eigenvalue (counting multiple eigenvalues separately) of the Laplacian matrix of G. [1] This eigenvalue is greater than 0 if and only if G is a connected graph. This is a corollary to the fact that the number ...
As a second-order differential operator, the Laplace operator maps C k functions to C k−2 functions for k ≥ 2.It is a linear operator Δ : C k (R n) → C k−2 (R n), or more generally, an operator Δ : C k (Ω) → C k−2 (Ω) for any open set Ω ⊆ R n.
where the degree of a vertex counts the number of times an edge terminates at that vertex. In an undirected graph , this means that each loop increases the degree of a vertex by two. In a directed graph , the term degree may refer either to indegree (the number of incoming edges at each vertex) or outdegree (the number of outgoing edges at ...
The connection Laplacian, also known as the rough Laplacian, is a differential operator acting on the various tensor bundles of a manifold, defined in terms of a Riemannian- or pseudo-Riemannian metric. When applied to functions (i.e. tensors of rank 0), the connection Laplacian is often called the Laplace–Beltrami operator.
In the mathematical field of graph theory, Kirchhoff's theorem or Kirchhoff's matrix tree theorem named after Gustav Kirchhoff is a theorem about the number of spanning trees in a graph, showing that this number can be computed in polynomial time from the determinant of a submatrix of the graph's Laplacian matrix; specifically, the number is equal to any cofactor of the Laplacian matrix.
The smallest pair of cospectral mates is {K 1,4, C 4 ∪ K 1}, comprising the 5-vertex star and the graph union of the 4-vertex cycle and the single-vertex graph. [1] The first example of cospectral graphs was reported by Collatz and Sinogowitz [2] in 1957. The smallest pair of polyhedral cospectral mates are enneahedra with eight vertices each ...