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      7
      votes
      0answers
      156 views

      Matrix of high rank mod $2$: must it have a large non-singular minor (with disjoint rows and columns)?

      Let $A$ be a $2n$-by-$2n$ matrix with entries in $\mathbb{Z}/2\mathbb{Z}$ such that, for every $2n$-by-$2n$ diagonal matrix $D$ with entries in $\mathbb{Z}/2\mathbb{Z}$, the matrix $A+D$ has rank $\...
      1
      vote
      1answer
      47 views

      Rank and edges in a combinatorial graph?

      Fix a $d\in\mathbb N$ and consider the matrix $M\in\{0,1\}^{2^d\times d}$ of all $0/1$ vectors of length $d$. Consider the matrix $G\in\{0,1\}^{n\times n}$ whose $ij$ the entry is $0$ if inner product ...
      0
      votes
      1answer
      68 views

      Energy of a symmetric matrix with $0$, $1$ or $-1$ entries

      I have a symmetric matrix with entries $0$, $1$ or $-1$ which appeared in my works in graph theory (the diagonal entries are all zero). I need a good upper bound for the energy of this matrix; i.e. "...
      2
      votes
      1answer
      159 views

      Eigenvalues of random graphs

      At time $t=0$, let $G_n(V,E)$ be a graph with $n$ vertices and $m < n$ edges. Then there exists a unique symmetric adjacency matrix $A_n$ associated with $G_n(V,E)$, defined as follows: $a_{ij} = 1$...
      2
      votes
      2answers
      140 views

      What's the full assumption for Laplacian matrix $L=BB^T=\Delta-A$?

      Graph with no-selfloop, no-multi-edges, unweighted. directed For directed graph Adjacency matrix is a non-symmetric matrix $A_{in}$ considering indegree or $A_{out}$ considering outdegree. Degree ...
      9
      votes
      0answers
      228 views
      +50

      Correspondence between matrix multiplication and a graph operation of Lovasz

      In his book "Large networks and graph limits" (available online here: http://web.cs.elte.hu/~lovasz/bookxx/hombook-almost.final.pdf), Lovasz describes a multiplication operation (he calls it ...
      2
      votes
      2answers
      150 views

      Adjacency matrix of total graph

      Is there a nice way of relating the adjacency, incidence , Laplacian matrices and other matrices associated to a graph of a total graph with its original graph, or, say, at least relating that of the ...
      5
      votes
      1answer
      291 views

      Determining the primitive order of a binary matrix

      Let ${\bf A}_n$ be an $2n \times 2n$ matrix that is defined as follows $$ {\bf A}_n=\left( \begin{array}{c} 0&0&\cdots&0&0&0&0&1&1\\ 0&0&\cdots&0&0&...
      0
      votes
      1answer
      53 views

      Estimating Maximal-Clique of Metric Graphs via the Rank of their Adjacency Matrix

      Let $\mathrm{M}\in\lbrace0,1\rbrace^n$ be the adjacency matrix of a graph $\mathrm{G}\left(V,E\subseteq\lbrace\lbrace u,v\rbrace| u,v\in V\rbrace\right)$ of order $n$. Let $\mathrm{G}$ ...
      1
      vote
      0answers
      62 views

      matching two positive-semidefinite matrices

      Let $M_1$ and $M_2$ be two real positive-semidefinite matrices. Is there any algorithm to compute a permutation matrix $P$ to minimize $\| M_1-PM_2P^T \|_F^2$ or equivalently to maximize $trace(...
      2
      votes
      0answers
      145 views

      Space of change of basis matrices between two similar matrices - how to reduce it with additional tests?

      Assume we have two real symmetric $n\times n$ matrices: $A, B$. We can easily test their similarity: $\textrm{Tr}(A^k)=\textrm{Tr}(B^k)$ for $k=1..n$. In this case both can be rotated to the same ...
      6
      votes
      1answer
      239 views

      Which zero-diagonal matrices contain the all-one vector in their columns' conic hull?

      Let $A$ be a non-negative zero-diagonal invertible matrix. Which $A$ make the following assertions true, which are all equivalent: The all-one vector $j$ is contained in the conic hull of $col(A)$. ...
      1
      vote
      1answer
      113 views

      Walks of odd Lengths in a Matrix

      Consider the following matrix $$ A=\left[ \begin {array}{cccc} 1&1&0&0\\ 0&0&1&0\\ 0&0&1&1\\ 1&0&0&0 \end {array} \right]. $$ Assume that $B=A^k$ ...
      2
      votes
      1answer
      139 views

      When does a row standardized adjacency matrix have a real spectrum?

      A colleague in spatial statistics was looking at a map with about 600 regions. For the application she's considering, the induced adjacency matrix had some undesirable properties (where two regions ...
      0
      votes
      0answers
      97 views

      The algebraic connectivity of $P_n$, the path on $n$ vertices does not exceed $\frac{12}{n^2-1}$

      The algebraic connectivity of $P_n$, the path on $n$ vertices does not exceed $\frac{12}{n^2-1}$. Let algebraic connectivity of $P_n$ be denoted by $\mu$. I have proved a result that if $G$ is a ...

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