K Shahul Hameed - Department of Mathematics, K M M Government Women’s College, Kannur, P.O. Box ۶۷۰۰۰۴, Kerala, India
K O Ramakrishnan - Department of Mathematics, K M M Government Women’s College, Kannur, P.O. Box ۶۷۰۰۰۴, Kerala, India

Given a matrix A with elements from a field of characteristic zero, the transin-verse A# of A is defined as the transpose of the matrix obtained by replacing the non-zero elements of A by their inverses and leaving zeros, if any, unchanged.We discuss the properties of this matrix operation in some detail and as an important application, we reinvent the celebrated matrix tree theorem for gain graphs.Characterization of balance in connected gain graphs using its Laplacian matrix becomes an immediate consequence.

Gain graph, Signed graph, graph eigenvalues, Graph Laplacian