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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 $\... 1answer 47 views ### Rank and edges in a combinatorial graph? Fix a$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 ... 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. "... 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:$a_{ij} = 1$... 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 ... 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 ... 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 ... 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 $${\bf A}_n=\left( \begin{array}{c} 0&0&\cdots&0&0&0&0&1&1\\ 0&0&\cdots&0&0&... 1answer 53 views ### Estimating Maximal-Clique of Metric Graphs via the Rank of their Adjacency Matrix Let \mathrm{M}\in\lbrace0,1\rbrace^n be the adjacency matrix of a graph \mathrm{G}\left(V,E\subseteq\lbrace\lbrace u,v\rbrace| u,v\in V\rbrace\right) of order n. Let \mathrm{G} ... 0answers 62 views ### matching two positive-semidefinite matrices Let M_1 and M_2 be two real positive-semidefinite matrices. Is there any algorithm to compute a permutation matrix P to minimize \| M_1-PM_2P^T \|_F^2 or equivalently to maximize trace(... 0answers 145 views ### Space of change of basis matrices between two similar matrices - how to reduce it with additional tests? Assume we have two real symmetric n\times n matrices: A, B. We can easily test their similarity: \textrm{Tr}(A^k)=\textrm{Tr}(B^k) for k=1..n. In this case both can be rotated to the same ... 1answer 239 views ### Which zero-diagonal matrices contain the all-one vector in their columns' conic hull? Let A be a non-negative zero-diagonal invertible matrix. Which A make the following assertions true, which are all equivalent: The all-one vector j is contained in the conic hull of col(A). ... 1answer 113 views ### Walks of odd Lengths in a Matrix Consider the following matrix$$ 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$B=A^k$... 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 ... 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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