For a quadratic matrix polynomial associated with a damped mass-spring system there are three types of critical eigenvalues, the eigenvalues $\infty$ and $0$ and the eigenvalues on the imaginary axis. All these are on the boundary of the set of (robustly) stable eigenvalues. For numerical methods, but also for (robust) stability analysis, it is desirable to deflate such eigenvalues by projecting the matrix polynomial to a lower dimensional subspace before computing the other eigenvalues and eigenvectors. We describe structure-preserving deflation strategies that deflate these eigenvalues via a trimmed structure-preserving linearization. We employ these results for the special case of hyperbolic problems. We also analyze the effect of a (possibly low rank) parametric damping matrix on purely imaginary eigenvalues.

A matrix $A$ is said to be an integer matrix if all its entries are integers. We prove that a nonsingular integer matrix $A$ has integer eigenvalues if and only if $A^{-1}$ can be written as the sum of $n$ rank-one matrices that meet certain requirements. A method for constructing integer matrices with integer eigenvalues using the Hadamard product is also provided. Let $S$ represent an $n$-tuple of nonnegative integers. If there is an $n\times n$ integer matrix $A$ whose spectrum is $S$, we say that $S$ is realisable by an integer matrix. In [On graphs whose Laplacian matrices have distinct integer eigenvalues, Journal of Graph Theory, 50(2):162–174, (2005)], the Fallat et al. posed a conjecture that “there is no simple graph on $n\geq 2$ vertices whose Laplacian spectrum is given by $(0, 1,\ldots,n-1)$.” We provide a characterization of threshold graphs using the spectra of quotient matrices of its $G$-join graphs. As a consequence, we prove that given any $n-1$ positive integers $\lambda _{2},\ldots ,\lambda _{n}$ such that $\lambda _{2}\leq \cdots \leq \lambda _{n}$, the $n$-tuple $(0,\lambda _{2},\ldots ,\lambda _{n})$ is realizable by the Laplacian matrix of a multidigraph. In particular, we show that $(0, 1,\ldots ,n-1)$ can be realizable by the Laplacian matrix of a multidigraph. This is a joint work with Subhasish Behera.

The graph of pyramids Tn,k (k>1) is a collection of n-k pyramids, each of which is a complete graph of k+1 vertices, sharing Kk as a common base. The Turan graph T(n,k) is a complete k-partite graph with n vertices where the sizes of the sizes of the parts are as equal as possible.

In the talk we will prove that the graphs of pyramids are determined by the spectrum of their adjacency matrix and give a new proof of a similar result for the Turan graphs.

The talk is based on joint work with Igal Sason, Noam Krupnik and Suleiman Hamud.

Recent advancements in multi-assay single-cell omics technologies have made it possible to collect multiple layers of molecular information from the same individual cells. Each omics modality—such as transcriptomics, epigenomics, or proteomics—offers a distinct perspective on cellular identity and function. Integrating these diverse data types holds significant potential for uncovering deeper and more comprehensive insights into complex cellular processes. However, modeling single-cell multi-omics data presents substantial challenges due to its high dimensionality, inherent sparsity, and technical noise.

To address these challenges, we introduce a novel integrative analytical framework “joint graph-regularized Single-Cell Kullback-Leibler Sparse Non-negative Matrix Factorization (jrSiCKLSNMF)—pronounced “junior sickles NMF.” This method is specifically designed to extract biologically meaningful latent factors that are shared across multiple omics modalities within the same set of single cells. By incorporating graph-based regularization, our approach leverages known relationships among cells to preserve local manifold structures, enhancing the interpretability and biological relevance of the learned factors.

Our implementation includes an efficient multiplicative update scheme that allows scalable computation even for large-scale datasets. We rigorously evaluated jrSiCKLSNMF on synthetic datasets generated using third-party simulation tools. Across a range of benchmarking scenarios, our method consistently outperformed several state-of-the-art approaches in terms of clustering accuracy and robustness.

Moreover, when applied to a real-world multi-assay single-cell omics dataset, jrSiCKLSNMF achieved clustering results that aligned well with known biological annotations, demonstrating its potential utility in practical applications. The method’s ability to integrate heterogeneous omics layers while maintaining sensitivity to subtle cell-type differences makes it a valuable tool for advancing our understanding of cellular heterogeneity in health and disease.

Domination problems form a central theme in graph theory, with applications ranging from communication networks to decision-making systems. In particular, the notion of $P_3$-convex domination captures interesting structural properties based on paths of length two. This talk presents a linear programming perspective to model and study such domination problems. We outline integer linear programming formulations that incorporate convexity constraints in a natural way. Relaxations of these formulations yield useful bounds and algorithmic insights. The approach provides an algorithm for integer linear programming (ILP) formulation of $P_3$-convex domination problem. Illustrative examples on graphs of varying classes will be discussed to highlight the methodology. Computational aspects and related challenges of $P_3$-convex domination will also be discussed.

A generalized inverse of $A$ is a matrix $X$ that satisfies $AXA = A$. Square matrices $A$ and $C$ of the same order are disjoint if the column spaces of $A$ and $C$ as well as row spaces are disjoint. $C$ is a complement of $A$ if $A$ and $C$ are disjoint and $A+C$ is nonsingular. Any g-inverse of $A$ can be obtained from $(A+C)^{-1}$ by suitably choosing complement $C$ of $A$. This method is called the Inverse rank Complemented Matrix (ICM) method. In the general Gauss-Markov model (GGM), $y = X\beta + \epsilon$ where $y$ and $\epsilon$ are vector random variables, $X$ is a design matrix, and $\sigma ^{2}G$ is a singular covariance matrix of the vector random variable $\epsilon$, $y$ can be split into four mutually uncorrelated random variables as $y = y_{1} + y_{2} +y_{3} +y_{4}$ by splitting $G$ into disjoint sections. Estimates of linear functions of $\beta$ under linear constraints with optimal properties follow as simple linear functions of the above four components. In this article, we demonstrate how various analytical issues widely discussed in the GGM literature can be resolved using the four components of $y$. We observe that the ICM method is a simple and direct method to analyze GGM as against the IPM method of inverting a partitioned matrix introduced by C. R. Rao.

Given a graph $G=(V,E)$ and a collection $\mathcal{C}$ of matrices associated with $G$, we consider maximizing the nullity over all matrices in $\mathcal{C}$. Colin de Verdi\`ere demonstrated that the maximum nullity associated with a certain collection of weighted Laplacian matrices is intimately related to the topology of a graph. In his work, for example, he developed an algebraic characterization of planar graphs! In this talk, we focus on the class, denoted by $S(G)$, of all real (square) symmetric matrices, $A=[a_{ij}]$ indexed by the vertices of $G$ in which for $i \neq j$, $a_{ij}\neq 0$ whenever $i ~ j$. For any symmetric matrix $n \times n$ $A$, we define the inertia of $A$ to be the triple ${\rm in}(A) = (p,q,v)$, where $p$ is the number of positive eigenvalues of $A$, $q$ is the number of negative eigenvalues of $A$ and $v=n-(p+q)$ is the nullity of $A$. For a graph $G$ on $n$ vertices, partition the set $S(G)$ into subsets $S_{q}(G) = \{ A \in S(G): {\rm in} (A)=(\cdot, q, \cdot)\}$ (i.e., the matrices in $S(G)$ with exactly $q$ negative eigenvalues), where $0 \leq q \leq n$. Define $M_q(G) = \max\{ {\rm nul}(A) : A \in S_q(G)\}$. In general terms, the main objective is to determine $M_q(G)$ for all such $q$ for a fixed graph or, more generally, a graph family.

In this lecture, we will review existing research on parameters $M_q$ for various families of graphs, including trees and threshold graphs. In addition, we will discuss the positive semi-definite case ($q=0$) or, equivalently, the minimum rank over all faithful orthogonal representations of a graph. finally, connections to zero forcing analogs and implications to the inverse inertia problem for graphs will also be highlighted.

The talk addresses research mathematicians and graduate students in algebra, combinatorics and functional analysis. The main objective is the study of algebras in which non-units or zero divisors can be expressed as a product of idempotents. This question originated in a 1966 paper by J. M. Howie who showed, among other things, that non-injective mappings from a finite set into itself are products of idempotents. This was followed in 1967 by J. A. Erdos who proved the analogous result for singular matrices over a field. In 1975, P .R. Halmos asked a question whether a singular matrix over a field can be expressed as a product of nilpotent matrices. This was answered by Sourour and later by R.P. Sullivan in 2008 as an application of the result that such matrices are product of idempotents. We say that an algebra R has the IPn property if each n ×n matrix over R with nonzero left and right annihilators, is a product of idempotent matrices. We say that an algebra has the IP property if it has IPn property for every n ≥ 2. Laffey’s proved that every Euclidean domain has the IP property. We have shown that a singular {0,1}-matrix over an integral domain of characteristic zero is a product of idempotents. It is also known that for commutative principal ideal domain the IP property is equivalent to the IP2 property. Moreover, for a right and left Bezout domain the IP2 property is equivalent to the IP property. We show that if a projective-free ring R has the IPn property for some n ≥ 1, then R is a domain. The question of representing a singular matrix over a von Neumann regular ring as a product of idempotents is related to a long-standing open question: whether there exists a von Neumann regular ring that is not separative. The conditions for representing a non-injective bounded linear operators on Banach spaces as products of idempotents are also obtained.

A right quaternion matrix polynomial is an expression of the form $P(\lambda) = \displaystyle \sum_{i=0}^{m}A_i \lambda^i$, where $A_i \in M_n(\mathbb{H})$ with $A_m \neq 0$, where $M_n(\mathbb{H})$ is the set of all square matrices from the ring of quaternions. A quaternion $\lambda_0 \in \mathbb{H}$ is a right eigenvalue of $P(\lambda)$ if there exists a nonzero vector $y \in \mathbb{H}^n$ such that $\displaystyle \sum_{i=0}^{m}A_i y\lambda_0^i =0$. The purpose of this talk is to bring out some recent results about the location of right eigenvalues of $P(\lambda)$ relative to certain subsets of the set of quaternions. The notion of (hyper)stability of complex matrix polynomials is extended to quaternion matrix polynomials and results are obtained about right eigenvalues of $P(\lambda)$ by $(1)$ giving a relation between (hyper)stability of a quaternion matrix polynomial and its complex adjoint matrix polynomial, and then by $(2)$ proving that $P(\lambda)$ is stable with respect to an open (closed) ball in the set of quaternions, centered at a complex number if and only if it is stable with respect to its intersection with the set of complex numbers. We derive as a consequence of the above that right eigenvalues of $P(\lambda)$ lie between two concentric balls of specific radii in the set of quaternions centered at the origin. A generalization of the Enestr{\”o}m-Kakeya theorem to quaternion matrix polynomials is obtained as an application. Finally, we also identify classes of quaternion matrix polynomials for which stability and hyperstability are equivalent. This talk is based on [1].

[1] Pallavi Basavaraju, Shrinath Hadimani and Sachindranath Jayaraman. Stability of quaternion matrix polynomials. Linear Algebra, Matrices and their Applications, Contemporary Mathematics, AMS, 2025, to appear.

A graph $G$ is said to be non-singular (resp. singular) if its adjacency matrix $A(G)$ is non-singular (resp. singular). Denote by $\mathcal{G}$ the class of all non-singular graphs $G$ for which $A(G)^{-1}$ has zero diagonal. We denote the weighted graph associated with $A(G)^{-1}$ by $G_+$. A graph $G$ in $\mathcal{G}$ is said to self invertible if $G$ is isomorphic to the underlying graph of $G_+$. It is well known that $T_+$ is a tree if and only if $T$ is a simple corona. Moreover, a simple corona is always self-invertible \cite{inv}. In this work, we extend these results to bicyclic graphs, giving a complete characterization of self-invertible bicyclic graphs. A graph $G$ is said to have SR-property if $\frac{1}{\lambda}$ is an eigenvalue of $A(G)$ whenever $\lambda$ is an eigenvalue of $A(G)$ with the same multiplicity. By exploiting the close relationship between self-invertibility and the SR-property, we obtain a complete classification of bicyclic graphs satisfying the SR-property.

Given an undirected graph $G$, we consider the continuous time quantum walk on $G$, which is governed by the transition matrix $U(t)=exp(itM), t\in \mathbb{R},$ where $M$ is a real symmetric matrix such that $m_{j,k}=0$ if and only if ${j,k}$ is not an edge of $G$. A key task in quantum computing is the accurate transfer of one quantum state to another via such a continuous time quantum walk.

Pure states correspond to one-dimensional subspaces of $\mathbb{C}^n$, and are represented by unit vectors. We develop the theory of perfect state transfer (PST) between real pure states, with emphasis on the adjacency and Laplacian matrices as Hamiltonians of a graph representing a quantum spin network. We characterize PST between real pure states based on the spectral information of the Hamiltonian, and prove three fundamental results: (i) every periodic real pure state $x$ admits PST with another real pure state $y$, (ii) every connected graph admits PST between real pure states, and (iii) given any pair of real pure states $x$ and $y$ and any time $\tau$, there exists a real symmetric matrix $M$ such that $x$ and $y$ admit PST relative to $M$ at time $\tau$. We also characterize the graphs, and the associated real pure states, having the minimum possible PST time, for the adjacency and Laplacian matrix cases.

Joint work with Chris Godsil and Hermie Monterde.

A pseudo linear map is the analogue of a linear map but in the setting of a ring with a derivation or a σ-derivation. The definition was first given by N. Jacobson (1937) a few years after the formal introduction of skew polynomial rings by O. Ore (1937). We can see these maps in the frame of representation (algebras, groups, Hopf algebras). We will analyze the similarities and the differences between pseudo linear maps and linear maps. Many natural analogues of linear classical topics will be mentioned. The strong connections between pseudo linear maps and evaluation of skew polynomials and some of its consequences will be given. We will also cover the case of different notions of evaluations for multivariate skew polynomials and show that they are connected with pseudo multilinear maps. These will be applied to answer an open question in a much more general frame than the original setting.

We explore the existence of pure Nash equilibria in bimatrix games, that are equilibria where each player deterministically plays one pure strategy.

Every finite noncooperative two-player nonzero-sum game, also called as a bimatrix game, has at least one Nash equilibrium in the mixed strategy extension, i.e. when players may play probability distributions over their available strategies. Special classes of games have been studied where a pure Nash equilibrium is guaranteed to exist, for example, subclasses of symmetric games, potential games ([2]), symmetric potential games and aggregative games.
Shapley ([3]) showed that a matrix game, namely a two-player zero sum game, has a pure saddle point if every 2$\times$2 subgame has one. For bimatrix games, however, a similar condition on 2$\times$2 subgames is not sufficient for the existence of pure Nash equilibria.

We prove the existence of pure Nash equilibria for symmetric bimatrix games that have the quasi harmonicity property ([1]) and quasiconcavity at the diagonal property. We also show existence of pure Nash equilibria for some classes of bimatrix games, not necessarily symmetric, by generalizing the results of Shapley. Some illustrative examples and directions for future work are discussed as well.

References

[1] L. Mallozzi and A. Sacco, Stackelberg-Nash equilibrium and quasi harmonic games, Annals of Operations Research, 318, 1029–1041, 2022.

[2] D. Monderer and L. S. Shapley, Potential games, Games and Economic Behavior, 14, 124–143, 1996.

[3] L. S. Shapley, Some topics in two-person games. In: Dresher, M., Shapley, L. S. and Tucker, A. W. (eds), Advances in game theory, Princeton University Press, Princeton, 1–28, 1964.

The Hermitian adjacency matrices of digraphs based on the sixth root of unity were introduced in [B. Mohar, A new kind of Hermitian matrices for digraphs, Linear Alg. Appl. (2020)]. They appear as the most natural choice for the spectral theory of digraphs. For undirected graphs, the spectrum of the adjacency matrix is symmetric around 0 if and only if the graph is bipartite. However, in the directed case, this is no longer true. There are orientations of nonbipartite graphs which are spectrally symmetric. Our main result concerns the extremal problem of maximizing the density of spectrally symmetric oriented graphs. The maximum possible density is shown to be between 13/18 and 10/11. This is joint work with Saieed Akbari, Jonathan Aloni, Maxwell Levit, and Steven Xia.

Copositivity has gained popularity in last three decades as a key concept for optimization, which can handle nonconvex, mixed-integer, linear complementarity problem and polynomial optimization problems. Copositivity plays a role in quadratic optimization, where the set of
copositive matrices can be used to obtain relaxations on the unknown optimal value. The notion of copositivity of a matrix is well known in the area of linear complementarity problems (LCP) in the context of existence results and results on the successful termination of Lemke’s algorithm. In the last decades there has been an increasing interest in this property of a matrix, and in linear optimization problems over the cone of copositive matrices. Several articles about the properties of the set of copositive matrices are proposed in the last sixty years. Much less is known about the copositive plus matrices, fully copositive matrices which form a subset of the copositive matrices. We discuss various open problems related to these classes. Further we discuss almost copositive matrices, almost fully copositive matrices, copositive matrices of exact order k and future direction of research.

In 1971, in his seminal paper [1], entitled Unified theory of linear estimation, C.R. Rao considered the properties of best linear unbiased estimators, BLUEs, in the general linear model $M(V) = \{ y, X\beta, V \}$, where $V$ refers to the covariance matrix of the observable random vector $y$ and $X$ is the model matrix. Both $X$ and the nonnegative definite covariance matrix $V$ are known. Citing Rao, “In Section 5 [of his paper] we raise the question of identification of $V$ given the class of BLUE’s of all estimable functions”. It is precisely Section 5 of Rao’s paper which is in our focus. In particular, we will take a good look at Rao’s Theorems 5.2 and 5.3 which answer the following question: Given the model $M(V_{0}) = \{y, X\beta, V_{0} \}$, how to characterize the set of all covariance matrices $V$ such that every representation of the BLUE of $X\beta$ under $M(V_{0})$ remains BLUE under $M(V)$. Our attempt is to provide some new insight into this problem area.

This talk is based on joint work with Stephen J. Haslett, Jarkko Isotalo and Augustyn Markiewicz.

[1] C. R. Rao. {\em Unified theory of linear estimation}. Sankhya Ser.~A, 33:371–394, 1971.

Scaling problems appear in military budget allocation, Estimation of Markov Transition probabilities that are doubly stochastic, improving accuracy in Gaussian elimination, Scaling in transportation planning via gravity models, arriving at maximum likelihood estimates via scaling in discrete multivarite statistical analysis. etc.The following existence theorem (R.B. Bapat and TES Raghavan (1989), J Lin Alg Appl , 114/115: 705-715) is central to this study:

Theorem: Let $K $ be a bounded non-empty polyhedron given by $$K = {π ∈ R^n : π ≥ 0, Cπ = b},$$ where $C = (c_{ij})$ is an $ m × n $ matrix and $ b ∈ R^m$ is a non-zero vector. Let $y ∈ K$. Then for any $x ≥ 0$ with the same zero pattern as $y$, there exist $ z_i > 0, i = 1, . . . ,m$ and there exists $ π ∈ K $ such that $$π_j = x_j \prod_{i=1}^m {z_{i}}^c_{ij}\quad i=1,\ldots m, j=1,2.\ldots n$$

Quadratic subspaces are useful, under a normality assumption, when estimating parameters in linearly structured covariance matrices when also their inverses are linearly structured. Different kind of applications will be considered.

There are many interesting results concerning the existence and uniqueness in the theory of the Linear Complementarity Problem (LCP) that are related to some notion of nonnegativity or nonpositivity of the matrices concerned. In this talk, first, I will present a survey of some of these results. This will be followed by a discussion of some new results obtained in this direction.

A graph G is a split graph, if its vertex set can be partitioned into an independent set and a clique. It is known that the diameter of such graphs is at most 3. Here, we shall first discuss a complete classification of the connected bidegreed 3-extremal split graphs and report some recent results. Also, we construct certain families of non-bidegreed 3-extremal split graphs.

In the second part, we obtain sharp bounds for the distance spectral radius of split graphs. We find the distance spectral radius of all biregular split graphs of diameter 2 and some biregular split graphs of diameter 3.

This is a joint work with Felix Goldberg, Steve Kirkland and Anu Varghese.