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The eccentricity matrix ɛ(G) of a graph G is obtained from the distance matrix of G by retaining the largest distances in each row and each column, and leaving zeros in the remaining ones. The eccentricity energy of G is sum of the absolute values of the eigenvalues of ɛ(G). Although the eccentricity matrices of graphs are closely related to the distance matrices of graphs, a number of properties of eccentricity matrices are substantially different from those of the distance matrices. The change in eccentricity energy of a graph due to an edge deletion is one such property. In this article, we give examples of graphs for which the eccentricity energy increase (resp., decrease) but the distance energy decrease (resp., increase) due to an edge deletion. Also, we prove that the eccentricity energy of the complete k-partite graph Kn 1, ... , nk with k ≥ 2 and ni ≥ 2, increases due to an edge deletion.

A complex unit gain graph ($\mathbb{T}$-gain graph), $\Phi = (G, \varphi)$ is a graph where the function $\varphi$ assigns a unit complex number to each orientation of an edge of $G$, and its inverse is assigned to the opposite orientation. A $ \mathbb{T} $-gain graph $\Phi$ is balanced if the product of the edge gains of each cycle (with a fixed orientation) is $1$. Signed graphs are special cases of $\mathbb{T}$-gain graphs. The adjacency matrix of $\Phi$, denoted by $ \mathbf{A}(\Phi)$ is defined canonically. The gain Laplacian for $\Phi$ is defined as $\mathbf{L}(\Phi) = \mathbf{D}(\Phi) - \mathbf{A}(\Phi)$, where $\mathbf{D}(\Phi)$ is the diagonal matrix with diagonal entries are the degrees of the vertices of $G$. The minimum number of vertices (resp., edges) to be deleted from $\Phi$ in order to get a balanced gain graph the frustration number (resp, frustration index). We show that frustration number and frustration index are bounded below by the smallest eigenvalue of $\mathbf{L}(\Phi)$. We provide several lower and upper bounds for extremal eigenvalues of $\mathbf{L}(\Phi)$ in terms of different graph parameters such as the number of edges, vertex degrees, and average $2$-degrees. The signed graphs are particular cases of the $\mathbb{T}$-gain graphs, all the bounds established in paper hold for signed graphs. Most of the bounds established here are new for signed graphs. Finally, we perform comparative analysis for all the obtained bounds in the paper with the state-of-the-art bounds available in the literature for randomly generated Erd\H{o}s-Re\'yni graphs. Some of the major highlights of our paper are the gain-dependent bounds, limit convergence of the bounds to the extremal eigenvalues, and optimal extremal bounds obtained by posing optimization problems to achieve the best possible bounds., Comment: Preliminary version. 32 pages

Let G be a simple, connected graph and let A(G) be the adjacency matrix of G. If D(G) is the diagonal matrix of the vertex degrees of G, then for every real $$\alpha \in [0,1]$$ , the matrix $$A_{\alpha }(G)$$ is defined as $$A_{\alpha }(G) = \alpha D(G) + (1- \alpha ) A(G).$$ The eigenvalues of the matrix $$A_{\alpha }(G)$$ form the $$A_{\alpha }$$ -spectrum of G. Let $$G_1 {\dot{\vee }} G_2$$ , $$G_1 {\underline{\vee }} G_2$$ , $$G_1 \langle \text {v} \rangle G_2$$ and $$G_1 \langle \text {e} \rangle G_2$$ denote the subdivision-vertex join, subdivision-edge join, R-vertex join and R-edge join of two graphs $$G_1$$ and $$G_2$$ , respectively. In this paper, we compute the $$A_{\alpha }$$ -spectra of $$G_1 {\dot{\vee }} G_2$$ , $$G_1 {\underline{\vee }} G_2$$ , $$G_1 \langle \text {v} \rangle G_2$$ and $$G_1 \langle \text {e} \rangle G_2$$ for a regular graph $$G_1$$ and an arbitrary graph $$G_2$$ in terms of their $$A_{\alpha }$$ -eigenvalues. As an application of these results, we construct infinitely many pairs of $$A_{\alpha }$$ -cospectral graphs.

We establish a characterization of almost P-matrices via a sign non-reversal property. In this we are inspired by the analogous results for N-matrices. Next, the interval hull of two m × n matrices A = ( a i j ) and B = ( b i j ) , denoted by I ( A , B ) , is the collection of all matrices C ∈ R n × n such that each c i j is a convex combination of a i j and b i j . Using the sign non-reversal property, we identify a finite subset of I ( A , B ) that determines if all matrices in I ( A , B ) are N-matrices/almost P-matrices. This provides a test for an entire class of matrices simultaneously to be N-matrices/almost P-matrices. We also establish analogous results for semipositive and minimally semipositive matrices. These characterizations may be considered similar in spirit to that of P-matrices by Bialas–Garloff [1] and Rohn–Rex [16] , and of positive definite matrices by Rohn [15] .

It is well known that the set of all $n \times n$ matrices with distinct eigenvalues is a dense subset of the set of all real or complex $n \times n$ matrices. In [D.J. Hartfiel. Dense sets of diagonalizable matrices. {\em Proceedings of the American Mathematical Society}, 123(6):1669--1672, 1995.], the author established a necessary and sufficient condition for a subspace of the set of all $n \times n$ matrices to have a dense subset of matrices with distinct eigenvalues. The objective of this article is to identify necessary and sufficient conditions for a subset of the set of all $n \times n$ real or complex matrices to have a dense subset of matrices with distinct eigenvalues. Some results of Hartfiel are extended, and the existence of dense subsets of matrices with distinct singular values in the subsets of the set of all real or complex matrices is studied. Furthermore, for a matrix function $F(x)$, defined on a closed and bounded interval whose entries are analytic functions, it is proved that the set of all points for which the matrix $F(x)$ has repeated analytic eigenvalues/analytic singular values is either a finite set or the whole domain of the function $F$.

In [A note about cospectral graphs for the adjacency and normalized Laplacian matrices. Linear Multilinear Algebra. 2010;58(3-4):387–390], Butler constructed a family of bipartite graphs, which are...

The distance matrix of a simple connected graph G is D(G)=(dij), where dij is the distance between the vertices i and j in G. We consider a weighted tree T on n vertices with edge weights are squar...

In this article, we construct bipartite graphs which are cospectral for both the adjacency and normalized Laplacian matrices using partitioned tensor product. This extends the construction of Ji, Gong, and Wang \cite{ji-gong-wang}. Our proof of the cospectrality of adjacency matrices simplifies the proof of the bipartite case of Godsil and McKay's construction \cite{godsil-mckay-1976}, and shows that the corresponding normalized Laplacian matrices are also cospectral. We partially characterize the isomorphism in Godsil and McKay's construction, and generalize Ji et al.'s characterization of the isomorphism to biregular bipartite graphs. The essential idea in characterizing the isomorphism uses Hammack's cancellation law as opposed to Hall's marriage theorem used by Ji et al., Comment: 13 pages

A matrix $A$ is totally positive (or non-negative) of order $k$, denoted $TP_k$ (or $TN_k$), if all minors of size $\leq k$ 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 $TP_k$ 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 $\mathbb{R}^k$ with alternating signs. We also show a similar characterization for all $TN_k$ 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 $TP$ matrices by Gantmacher-Krein [Compos. Math. 1937] and $P$-matrices by Gale-Nikaido [Math. Ann. 1965]. As an application, we study the interval hull $\mathbb{I}(A,B)$ of two $m \times n$ matrices $A=(a_{ij})$ and $B = (b_{ij})$. This is the collection of $C \in \mathbb{R}^{m \times n}$ such that each $c_{ij}$ is between $a_{ij}$ and $b_{ij}$. Using the sign non-reversal property, we identify a two-element subset of $\mathbb{I}(A,B)$ that detects the $TP_k$ property for all of $\mathbb{I}(A,B)$ for arbitrary $k \geq 1$. 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 $\mathbb{I}(A,B)$., Final version, to appear in Bulletin of the London Mathematical Society. 9 pages, no figures

In this article we introduce the notion of P-proper splitting for square matrices. For an inconsistent linear system of equations $$Ax =b$$ , we associate an iterative method based on a P-proper splitting of A, which if convergent, converges to the best least squares solution of this system. We extend a result of Stein, using which we prove that if A is positive semidefinite, then the said iterative method converges. Also, we generalize Sylvester’s law of inertia and as an application of this generalization we establish some properties of P-proper splittings. Finally, we prove a comparison theorem for iterative methods associated with P-proper splittings of a positive semidefinite matrix.

An SPN matrix is a matrix which is the sum of a real positive semidefinite matrix and a symmetric nonnegative one. We solve the SPN completion problem: we show that the SPN completable graphs are the graphs in which every odd cycle induces a complete subgraph.

A real square matrix A is called a P-matrix if all its principal minors are positive. Such a matrix can be characterized by the sign non-reversal property. Taking a cue from this, the notion of a P-operator is extended to infinite dimensional spaces as the first objective. Relationships between invertibility of some subsets of intervals of operators and certain P-operators are then established. These generalize the corresponding results in the matrix case. The inheritance of the property of a P-operator by the Schur complement and the principal pivot transform is also proved. If A is an invertible M-matrix, then there is a positive vector whose image under A is also positive. As the second goal, this and another result on intervals of M-matrices are generalized to operators over Banach spaces. Towards the third objective, the concept of a Q-operator is proposed, generalizing the well known Q-matrix property. An important result, which establishes connections between Q-operators and invertible M-op...

Coconut is inseparable part of life of people of southern India particularly in the states of Kerala and Tamilnadu. Coconut as tender coconut water, coconut gratings, coconut milk, coconut oil etc. find its way in at least one food item cooked daily the people of this southern part of India. Due to extreme shortage in people to climb the coconut trees and pluck the coconuts, the cost of coconuts is increasing steeply. One solution to this problem is to have a robotic coconut tree climber with an arm to cut the coconuts. In this research work, we present the design, implementation and testing of robotic arms to be used in these robotic coconut tree climbers, to cut the coconuts. Different robotic arm designs are presented with proper analysis and results.

H-matrices (matrices whose comparison matrix is an M-matrix) are well studied in matrix theory and have numerous applications, e.g., linear complementarity problems and iterative methods for solving systems of linear equations. In this article we establish some properties of their tensor counterparts: strong H -tensors and (general) H -tensors.

The resistance matrix of a simple connected graph G is denoted by R, and is defined by R = ( r i j ) , where r i j is the resistance distance between the vertices i and j of G. In this paper, we consider the resistance matrix of weighted graph with edge weights being positive definite matrices of same size. We derive a formula for the determinant and the inverse of the resistance matrix. Then, we establish an interlacing inequality for the eigenvalues of resistance and Laplacian matrices of tree. Using this interlacing inequality, we obtain the inertia of the resistance matrix of tree.

A complex unit gain graph is a simple graph in which each orientation of an edge is given a complex number with modulus 1 and its inverse is assigned to the opposite orientation of the edge. In this article, first we establish bounds for the eigenvalues of the complex unit gain graphs. Then we study some of the properties of the adjacency matrix of complex unit gain graph in connection with the characteristic and the permanental polynomials. Then we establish spectral properties of the adjacency matrices of complex unit gain graphs. In particular, using Perron-Frobenius theory, we establish a characterization for bipartite graphs in terms of the set of eigenvalues of gain graph and the set of eigenvalues of the underlying graph. Also, we derive an equivalent condition on the gain so that the eigenvalues of the gain graph and the eigenvalues of the underlying graph are the same., Comment: 14 pages, 2 figures

The generalized principal pivot transform is a generalization of the principal pivot transform to the singular case, using the Moore–Penrose inverse. In this article we study some properties of the generalized principal pivot transform. We prove that the Moore–Penrose inverse of a range-symmetric, almost skew-symmetric matrix is almost skew-symmetric. It is shown that the generalized principal pivot transform preserves the rank of the symmetric part of a matrix under some conditions.

In this article, we define new classes of tensors called double $\overline{B}$-tensors, quasi-double $\overline{B}$-tensors and establish some of their properties. Using these properties, we construct new regions viz., double $\overline{B}$-intervals and quasi-double $\overline{B}$-intervals, which contain all the $H$-eigenvalues of real even order symmetric tensors. We prove that the double $\overline{B}$-intervals is contained in the quasi-double $\overline{B}$-intervals and quasi-double ${\overline{B}}$-intervals provide supplement information on the Brauer-type eigenvalues inclusion set of tensors. These are analogous to the double $\overline{B}$-intervals of matrices established by J. M. Pe\~na~[On an alternative to Gerschgorin circles and ovals of Cassini, Numer. Math. 95 (2003), no. 2, 337-345.], Comment: Comments are welcome

For A, B ? ?n�n, let r(A,B) (c(A,B)) be the set of matrices whose rows, (columns) are independent convex combinations of the rows (columns) of A and B. Johnson and Tsatsomeros have shown that the set r(A,B) (c(A,B)) consists entirely of nonsingular matrices if and only if BA-1 (B-1A) is a P-matrix. For A,B ? ?n�n, let i(A, B) = {C ? ?n�n: min{aij, bij} ? cij ? max{aij, bij}}. Rohn has shown that if all the matrices in i(A, B) are invertible, then BA-1, A-1B, AB-1 and B-1A are P-matrices. In this article, we define a new class of matrices called P�-matrices and present certain extensions of the above results to the singular case, where the usual inverse is replaced by the Moore-Penrose generalized inverse. The case of the group inverse is briefly discussed. � 2013 � 2013 Taylor & Francis.

In this article we prove the strict monotonicity of the spectral radius of weakly irreducible nonnegative tensors. As an application, we give a necessary and sufficient condition for an interval hull of tensors to be contained in the set of all strong $\mathcal{M}$-tensors. We also establish some properties of $\mathcal{M}$-tensors. Finally, we consider some problems related to interval hull of positive (semi)definite tensors and $P(P_0)$-tensors., Comment: 10 pages

The cytochrome P 450 1A1 gene encoding a phase I metabolic enzyme appears to be a candidate for breast cancer risk. It is involved in the phase I detoxification of polycyclic aromatic hydrocarbons (PAHs) and 2-hydroxylation of estrogens and mammary carcinogens into 2-hydroxy catechol metabolites. Several studies have investigated polymorphisms in CYP1A1 and breast cancer risk with inconsistent results. We here carried out a population based case-control study of the CYP MspI (CYP1A1*1/M1) and Ile462Val (CYP1A1*2/M2) polymorphisms in CYP1A1 to clarify their importance in determining breast cancer susceptibility in a South Indian population. A total of 50 cases and 50 controls were genotyped for both polymorphisms. We also investigated putative interactions with exposure to pollution, radiation and intake of tobacco and CYP1A1 genotype and breast cancer risk using a case only study design. The genotype distribution of CYP1A1*1 in cancer patients was 6% for homozygous (CYP1A1 M1 [C/C], 34% for heterozygous CYP1A1 M1 [T/C] and 60% for wild type (CYP1A1 M1 [T/T] (OR: 0.583, CI-95% (0.252-1.348). The genotype distribution of M2 genotypes in patients was 24% of homozygous (CYP1A1 M2 [Val/Val], 4% for heterozygous (CYP1A1 M2 [Ile/Val] and 72% for wild type allele (CYP1A1 M2 [Ile/Ile] [OR: 0.720, CI-95% (0.606-0.856)]. Our results suggest that there is no significant correlation between CYP1A1 M1/ CYP1A1 M2 polymorphism and occurrence of breast cancer in South Indian women.

For A , B ∈ R m × n , let J = [ A , B ] be the set of all matrices C such that A ≤ C ≤ B , where the order is component wise. Krasnosel’skij et al. [9] and Rohn [11] have shown that if A and B are invertible with A - 1 ≥ 0 and B - 1 ≥ 0 , then every C ∈ J is invertible with C - 1 ≥ 0 . In this article, we present certain extensions of this result to the singular case, where the nonnegativity of the usual inverses is replaced by the nonnegativity of the Moore–Penrose inverse.