Type 2 Diabetes mellitus (T2DM) is characterized by peripheral insulin resistance, impaired insulin secretion, and reduced β-cell mass. Mechanisms that underlie β-cell failure include glucotoxicity, lipotoxicity, endoplasmic reticulum (ER) stress, and oxidative stress. This study was designed to assess the protective effect of trigonelline and diosgenin against changes in ER stress-associated apoptotic proteins CHOP, Caspase12, and Caspase3 and antioxidant levels in pancreas as well as adipose tissue PPARγ mRNA in T2DM rats. Markers of diabetes and obesity such as serum glucose, insulin, free fatty acid (FFA), TNF-α, IL-6, and leptin were also assessed. T2DM rats showed significantly elevated levels of pancreatic ER stress proteins and lipid peroxidation, while the antioxidants were significantly reduced. Histological examination also confirmed T2DM-associated damage in pancreas. In addition, a significant increase in serum FFA, TNF-α, IL-6, and decrease in leptin levels along with significantly decreased adipose mass and reduced PPARγ expression were observed in T2DM rats. On the other hand, trigonelline and diosgenin treatment independently brought about significant improvement in serum parameters, decrease in apoptotic ER stress proteins, and reinforced antioxidant status in pancreas. Histological examination of pancreas showed normal morphology. Treated groups also showed increased adipose tissue mass and enhanced PPARγ expression. Data from docking studies indicated good interaction of both compounds with PPARγ, and diosgenin showed better binding efficiency. These findings suggest that the insulin-sensitizing effects of trigonelline and diosgenin are mediated through moderation of ER stress and oxidative stress in pancreas as well as by PPARγ activation in adipose tissue.

The distance matrix of a simple connected graph G is D(G) = (d ij ), where d ij is the distance between the vertices i and j in G. We consider a weighted tree T on n vertices with edge weights are square matrix of same size. The distance d ij between the vertices i and j is the sum of the weight matrices of the edges in the unique path from i to j. In this article we establish a characterization for the trees in terms of rank of (matrix) weighted Laplacian matrix associated with it. Then we establish a necessary and sufficient condition for the distance matrix D, with matrix weights, to be invertible and the formula for the inverse of D, if it exists. Also we study some of the properties of the distance matrices of matrix weighted trees in connection with the Laplacian matrices, g-inverses and eigenvalues.

A matrix A is totally positive (or non-negative) of order k, denoted T P k (or T N k ), if all minors of size k are positive (or non-negative). It is well known that such matrices are characterized by the variation diminishing property together with the sign non-reversal property. We do away with the former, and show that A is T P k if and only if every submatrix formed from at most k consecutive rows and columns has the sign non-reversal property. In fact, this can be strengthened to only consider test vectors in R k with alternating signs. We also show a similar characterization for all T N k matrices -more strongly, both of these characterizations use a single vector (with alternating signs) for each square submatrix. These characterizations are novel, and similar in spirit to the fundamental results characterizing T P matrices by Gantmacher-Krein (Compos. Math. 4 (1937) 445-476) and P -matrices by Gale-Nikaido (Math. Ann. 159 (1965) 81-93).As an application, we study the interval hull I(A, B) of two m × n matrices A = (aij) and B = (bij). This is the collection of C ∈ R m×n such that each cij is between aij and bij. Using the sign non-reversal property, we identify a two-element subset of I(A, B) that detects the T P k property for all of I(A, B) for arbitrary k 1. In particular, this provides a test for total positivity (of any order), simultaneously for an entire class of rectangular matrices. In parallel, we also provide a finite set to test the total non-negativity (of any order) of an interval hull I(A, B).

A T-gain graph is a simple graph in which a unit complex number is assigned to each orientation of an edge, and its inverse is assigned to the opposite orientation. The associated adjacency matrix is defined canonically, and is called T-gain adjacency matrix. Let T G denote the collection of all T-gain adjacency matrices on a graph G. In this article, we study the cospectrality of matrices in T G and we establish equivalent conditions for a graph G to be a tree in terms of the spectrum and the spectral radius of matrices in T G . We identify a class of connected graphs F such that for each G ∈ F , the matrices in T G have nonnegative real part up to diagonal unitary similarity. Then we establish bounds for the spectral radius of T-gain adjacency matrices on G ∈ F in terms of their largest eigenvalues. Thereupon, we characterize T-gain graphs for which the spectral radius of the associated T-gain adjacency matrices equal to the largest vertex degree of the underlying graph. These bounds generalize results known for the spectral radius of Hermitian adjacency matrices of digraphs and provide an alternate proof of a result about the sharpness of the bound in terms of largest vertex degree established in [Krystal Guo, Bojan Mohar. Hermitian adjacency matrix of digraphs and mixed graphs.

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