The bipartite Laplacian matrix of a nonsingular tree

For a bipartite graph, the complete adjacency matrix is not necessary to display its adjacency information. In 1985, Godsil used a smaller size matrix to represent this, known as the bipartite adjacency matrix. Recently, the bipartite distance matrix of a tree with perfect matching was introduced as...

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Bibliographic Details
Published in:Special matrices 2023-08, Vol.11 (1), p.21-33
Main Authors: Bapat, Ravindra B., Jana, Rakesh, Pati, Sukanta
Format: Article
Language:English
Subjects:
05C05
05C12
05C50
15A15
bipartite distance matrix
bipartite Laplacian matrix
distance matrix
Laplacian matrix
tree
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Summary:For a bipartite graph, the complete adjacency matrix is not necessary to display its adjacency information. In 1985, Godsil used a smaller size matrix to represent this, known as the bipartite adjacency matrix. Recently, the bipartite distance matrix of a tree with perfect matching was introduced as a concept similar to the bipartite adjacency matrix. It has been observed that these matrices are nonsingular, and a combinatorial formula for their determinants has been derived. In this article, we provide a combinatorial description of the inverse of the bipartite distance matrix and establish identities similar to some well-known identities. The study leads us to an unexpected generalization of the usual Laplacian matrix of a tree. This generalized Laplacian matrix, which we call the , is usually not symmetric, but it shares many properties with the usual Laplacian matrix. In addition, we study some of the fundamental properties of the bipartite Laplacian matrix and compare them with those of the usual Laplacian matrix.
ISSN:2300-7451
2300-7451
DOI:10.1515/spma-2023-0102