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Keivan Hassani Monfared

Publications and source records attributed to Keivan Hassani Monfared.

10 recordsLinked to original sources
arXiv2019

Inertias of Laplacian matrices of weighted signed graphs

We study the sets of inertias achieved by Laplacian matrices of weighted signed graphs. First we characterize signed graphs with a unique Laplacian inertia. Then we show that there is a sufficiently small perturbation of the nonzero weights on the edges of any connected weighted signed graph so that all eigenvalues of its Laplacian matrix are simple. Next, we give upper bounds on the number of possible Laplacian inertias for signed graphs with a fixed flexibility $τ$ (a combinatorial parameter of signed graphs), and show that these bounds are sharp for an infinite family of signed graphs. Finally, we provide upper bounds for the number of possible Laplacian inertias of signed graphs in terms of the number of vertices.

math.SP↗
arXiv2018

Inverse Spectral Problems for Linked Vibrating Systems and Structured Matrix Polynomials

We show that for a given set $Λ$ of $nk$ distinct real numbers $λ_1, λ_2, \ldots, λ_{nk}$ and $k$ graphs on $n$ nodes, $G_0, G_1,\ldots,G_{k-1}$, there are real symmetric $n\times n$ matrices $A_s$, $s=0,1,\ldots, k$, such that the matrix polynomial $A(z) := A_k z^k + \cdots + A_1 z + A_0$ has $Λ$ as its spectrum, the graph of $A_s$ is $G_s$ for $s=0,1,\ldots,k-1$, and $A_k$ is an arbitrary positive definite diagonal matrix. When $k=2$, this solves a physically significant inverse eigenvalue problem for linked vibrating systems (see Corollary 5.3).

math.SP↗
arXiv2018

Community structure detection and evaluation during the pre- and post-ictal hippocampal depth recordings

Detecting and evaluating regions of brain under various circumstances is one of the most interesting topics in computational neuroscience. However, the majority of the studies on detecting communities of a functional connectivity network of the brain is done on networks obtained from coherency attributes, and not from correlation. This lack of studies, in part, is due to the fact that many common methods for clustering graphs require the nodes of the network to be `positively' linked together, a property that is guaranteed by a coherency matrix, by definition. However, correlation matrices reveal more information regarding how each pair of nodes are linked together. In this study, for the first time we simultaneously examine four inherently different network clustering methods (spectral, heuristic, and optimization methods) applied to the functional connectivity networks of the CA1 region of the hippocampus of an anaesthetized rat during pre-ictal and post-ictal states. The networks are obtained from correlation matrices, and its results are compared with the ones obtained by applying the same methods to coherency matrices. The correlation matrices show a much finer community structure compared to the coherency matrices. Furthermore, we examine the potential smoothing effect of choosing various window sizes for computing the correlation/coherency matrices.

cs.SI↗
arXiv2018

An Analog of Matrix Tree Theorem for Signless Laplacians

A spanning tree of a graph is a connected subgraph on all vertices with the minimum number of edges. The number of spanning trees in a graph $G$ is given by Matrix Tree Theorem in terms of principal minors of Laplacian matrix of $G$. We show a similar combinatorial interpretation for principal minors of signless Laplacian $Q$. We also prove that the number of odd cycles in $G$ is less than or equal to $\frac{\det(Q)}{4}$, where the equality holds if and only if $G$ is a bipartite graph or an odd-unicyclic graph.

math.CO↗
arXiv2016

A Structured Inverse Spectrum Problem For Infinite Graphs

It is shown that for a given infinite graph $G$ on countably many vertices, and a compact, infinite set of real numbers $Λ$ there is a real symmetric matrix $A$ whose graph is $G$ and its spectrum is $Λ$. Moreover, the set of limit points of $Λ$ equals the essential spectrum of $A$, and the isolated points of $Λ$ are eigenvalues of $A$ with multiplicity one. It is also shown that any two such matrices constructed by our method are approximately unitarily equivalent.

math.SP↗
arXiv2016

Existence of a Not Necessarily Symmetric Matrix with Given Distinct Eigenvalues and Graph

For given k distinct complex conjugate pairs, l distinct real numbers, and a given graph G on 2k+l vertices with a matching of size at least k, we will show that there is a real matrix whose eigenvalues are the given numbers and its graph is G. In particular, this implies that any real matrix with distinct eigenvalues is similar to a real, irreducible, tridiagonal matrix.

math.SP↗
arXiv2016

Spectral characterization of matchings in graphs

A spectral characterization of the matching number (the size of a maximum matching) of a graph is given. More precisely, it is shown that the graphs G of order n whose matching number is k are precisely those graphs with the maximum skew rank 2k such that for any given set of k distinct nonzero purely imaginary numbers there is a real skew-symmetric matrix A with graph G whose spectrum consists of the given k numbers, their conjugate pairs, and n-2k zeros.

math.CO↗
arXiv2015

On the Principal Permanent Rank Characteristic Sequences of Graphs and Digraphs

The principal permanent rank characteristic sequence is a binary sequence $r_0 r_1 \ldots r_n$ where $r_k = 1$ if there exists a principal square submatrix of size $k$ with nonzero permanent and $r_k = 0$ otherwise, and $r_0 = 1$ if there is a zero diagonal entry. A characterization is provided for all principal permanent rank sequences obtainable by the family of nonnegative matrices as well as the family of nonnegative symmetric matrices. Constructions for all realizable sequences are provided. Results for skew-symmetric matrices are also included.

math.CO↗
arXiv2014

Construction of real skew-symmetric matrices from interlaced spectral data and graph

A 1989 result of Duarte asserts that for a given tree T on n vertices, a fixed vertex i, and two sets of distinct real numbers L, M of sizes n and n-1, respectively, such that M strictly interlaces L, there is a real symmetric matrix A such that graph of A is T, eigenvalues of A are given by L, and eigenvalues of A(i) are given by M. In 2013, a similar result for connected graphs was published by Hassani Monfared and Shader, using the Jacobian method. Analogues of these results are presented here for real skew-symmetric matrices whose graphs belong to a certain family of trees, and all of their supergraphs.

math.CO↗