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Register a new userThe Determinant of ${pm 1}$-Matrices and Oriented Hypergraphs
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The determinants of ${pm 1}$-matrices are calculated by via the oriented hypergraphic Laplacian and summing over an incidence generalization of vertex cycle-covers. These cycle-covers are signed and partitioned into families based on their hyperedge containment. Every non-edge-monic family is shown to contribute a net value of $0$ to the Laplacian, while each edge-monic family is shown to sum to the absolute value of the determinant of the original incidence matrix. Simple symmetries are identified as well as their relationship to Hadamards maximum determinant problem. Finally, the entries of the incidence matrix are reclaimed using only the signs of an adjacency-minimal set of cycle-covers from an edge-monic family.
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For a given hypergraph, an orientation can be assigned to the vertex-edge incidences. This orientation is used to define the adjacency and Laplacian matrices. In addition to studying these matrices, several related structures are investigated including the incidence dual, the intersection graph (line graph), and the 2-section. A connection is then made between oriented hypergraphs and balanced incomplete block designs.
An oriented hypergraph is an oriented incidence structure that extends the concepts of signed graphs, balanced hypergraphs, and balanced matrices. We introduce hypergraphic structures and techniques that generalize the circuit classification of the signed graphic frame matroid to any oriented hypergraphic incidence matrix via its locally-signed-graphic substructure. To achieve this, Camions algorithm is applied to oriented hypergraphs to provide a generalization of reorientation sets and frustration that is only well-defined on balanceable oriented hypergraphs. A simple partial characterization of unbalanceable circuits extends the applications to representable matroids demonstrating that the difference between the Fano and non-Fano matroids is one of balance.
An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of $+1$ or $-1$. The adjacency and Laplacian eigenvalues of an oriented hypergraph are studied. Eigenvalue bounds for both the adjacency and Laplacian matrices of an oriented hypergraph which depend on structural parameters of the oriented hypergraph are found. An oriented hypergraph and its incidence dual are shown to have the same nonzero Laplacian eigenvalues. A family of oriented hypergraphs with uniformally labeled incidences is also studied. This family provides a hypergraphic generalization of the signless Laplacian of a graph and also suggests a natural way to define the adjacency and Laplacian matrices of a hypergraph. Some results presented generalize both graph and signed graph results to a hypergraphic setting.
An oriented hypergraph is an oriented incidence structure that extends the concept of a signed graph. We introduce hypergraphic structures and techniques central to the extension of the circuit classification of signed graphs to oriented hypergraphs. Oriented hypergraphs are further decomposed into three families -- balanced, balanceable, and unbalanceable -- and we obtain a complete classification of the balanced circuits of oriented hypergraphs.
Let $mathbb{F}_q$ be an arbitrary finite field of order $q$. In this article, we study $det S$ for certain types of subsets $S$ in the ring $M_2(mathbb F_q)$ of $2times 2$ matrices with entries in $mathbb F_q$. For $iin mathbb{F}_q$, let $D_i$ be the subset of $M_2(mathbb F_q)$ defined by $ D_i := {xin M_2(mathbb F_q): det(x)=i}.$ Then our results can be stated as follows. First of all, we show that when $E$ and $F$ are subsets of $D_i$ and $D_j$ for some $i, jin mathbb{F}_q^*$, respectively, we have $$det(E+F)=mathbb F_q,$$ whenever $|E||F|ge {15}^2q^4$, and then provide a concrete construction to show that our result is sharp. Next, as an application of the first result, we investigate a distribution of the determinants generated by the sum set $(Ecap D_i) + (Fcap D_j),$ when $E, F$ are subsets of the product type, i.e., $U_1times U_2subseteq mathbb F_q^2times mathbb F_q^2$ under the identification $ M_2(mathbb F_q)=mathbb F_q^2times mathbb F_q^2$. Lastly, as an extended version of the first result, we prove that if $E$ is a set in $D_i$ for $i e 0$ and $k$ is large enough, then we have [det(2kE):=det(underbrace{E + dots + E}_{2k~terms})supseteq mathbb{F}_q^*,] whenever the size of $E$ is close to $q^{frac{3}{2}}$. Moreover, we show that, in general, the threshold $q^{frac{3}{2}}$ is best possible. Our main method is based on the discrete Fourier analysis.
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