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Gain graphs are graphs where the edges are given some orientation and labeled with the elements (called gains) from a group so that gains are inverted when we reverse the direction of the edges. Generalizing the notion of gain graphs, skew gain graphs have the property that the gain of a reversed edge is the image of edge gain under an anti-involution. In this paper, we study two different types, Laplacian and g-Laplacian matrices for a skew gain graph where the skew gains are taken from the multiplicative group Fx of a field F of characteristic zero. Defining incidence matrix, we also prove the matrix tree theorem for skew gain graphs in the case of the g-Laplacian matrix.

A signed network Σ = (G , σ) with the underlying graph G = (V , E) , used as a mathematical model for analyzing social networks, has each edge in E with a weight 1 or − 1 assigned by the signature function σ. In this paper, we deal with two types of Detour Distance Laplacian (DDL) matrices for signed networks. We characterize balance in signed social networks using these matrices and we compute the DDL spectrum of certain unbalanced signed networks, as balanced signed networks behave like unsigned ones. [ABSTRACT FROM AUTHOR]

A conjugate skew gain graph is a skew gain graph with the labels (also called, the conjugate skew gains) from the field of complex numbers on the oriented edges such that they get conjugated when we reverse the orientation. In this paper, we introduce distance matrices for conjugate skew gain graphs and characterize balanced conjugate skew gain graphs using these matrices. We provide explicit formulae for the distance spectra of certain conjugate skew gain graphs. [ABSTRACT FROM AUTHOR]

In this paper, a simple automatic Immunohistochemical (IHC) scoring method based on H-score has been described in which, tissue sections are segmented using Otsu method and each cell nuclei is classified into one of five different categories by multi-level thresholding of feature extracted using maximal separation (MS) techniques and finally, total score (TS) and diagnostic index (DI) are determined. The performance of the automatic scoring method is evaluated by comparing with manual scores using correlation coefficient (Cc), mean absolute difference (MAD) and 95% confidence interval (CI). According to the performance evaluation, DI score based on MSBlue (blue component) had a strong agreement with both manual scores of tissue images. Therefore, automatic scoring based on MSBlue technique has high potential in modern medical practice of diagnosis, prognosis and targeted therapy design for cancer patients. [ABSTRACT FROM AUTHOR]

A gain graph is a graph where the edges are given some orientation and labeled with the elements (called gains) from a group so that gains are inverted when we reverse the direction of the edges. A signed graph with labels from the multiplicative group {1,–1} on the edges can be taken as a particular case of a gain graph. In this article, we extend the definition of certain operations on graphs, such as NEPS and composition, to gain graphs and characterize balance in such products of gain graphs. Also we analyze the adjacency matrices and eigenvalues of these products when the underlying group for labeling the edges of the gain graphs is the multiplicative group of a field. [ABSTRACT FROM AUTHOR]

Abstract: A gain graph is a graph where the edges are given some orientation and labeled with the elements (called gains) from a group so that gains are inverted when we reverse the direction of the edges. A signed graph with labels from the multiplicative group on the edges can be taken as a particular case of a gain graph. In this article, we initiate a matrix analysis of gain graphs by defining the adjacency matrices of gain graphs when the underlying group for labeling the edges is the multiplicative group of a field and characterize balance in such gain graphs using the characteristic polynomials. We also establish recurrence relations for the characteristic polynomials of a gain graph and discuss consequences of these relations. [Copyright &y& Elsevier]