Abstract
Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, (G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.
Highlights
The normalized distance Laplacian, DL(G) = I −T(G)− / D(G)T(G)− /, is introduced. This is analogous to the normalized Laplacian matrix, L(G) = I − D(G)− / A(G)D(G)− /, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix
If a weighted graph G has a set of two or more adjacent twins of degree d, dd+ is an eigenvalue of L(G) and − is an eigenvalue of A(G)
In this paper we introduced the normalized distance Laplacian
Summary
Spectral graph theory is the study of matrices de ned in terms of a graph, speci cally relating the eigenvalues of the matrix to properties of the graph. We nd bounds on the normalized distance Laplacian eigenvalues and provide data that leads to conjectures about the graphs achieving the maximum and minimum spectral radius. In the remainder of this section, we present results that are known to hold for the adjacency matrix and its Laplacians, followed by their generalizations to the distance matrices Both the normalized Laplacian and the normalized distance Laplacian include square roots (unless the graph is regular or transmission regular, respectively). [10] For all weighted connected graphs G, = μ < μ ≤ · · · ≤ μn ≤ , with μn = if and only if G is non-trivial and bipartite This result generalizes with one notable di erence: The normalized distance Laplacian never achieves 2 as an eigenvalue for n ≥.
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