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Full-Text Articles in Mathematics
Summing Cubes By Counting Rectangles, Arthur T. Benjamin, Jennifer J. Quinn, Calyssa Wurtz
Summing Cubes By Counting Rectangles, Arthur T. Benjamin, Jennifer J. Quinn, Calyssa Wurtz
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No abstract provided in this article.
Self-Avoiding Walks And Fibonacci Numbers, Arthur T. Benjamin
Self-Avoiding Walks And Fibonacci Numbers, Arthur T. Benjamin
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By combinatorial arguments, we prove that the number of self-avoiding walks on the strip {0, 1} × Z is 8Fn − 4 when n is odd and is 8Fn − n when n is even. Also, when backwards moves are prohibited, we derive simple expressions for the number of length n self-avoiding walks on {0, 1} × Z, Z × Z, the triangular lattice, and the cubic lattice.
The Linear Complexity Of A Graph, David L. Neel, Michael E. Orrison Jr.
The Linear Complexity Of A Graph, David L. Neel, Michael E. Orrison Jr.
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The linear complexity of a matrix is a measure of the number of additions, subtractions, and scalar multiplications required to multiply that matrix and an arbitrary vector. In this paper, we define the linear complexity of a graph to be the linear complexity of any one of its associated adjacency matrices. We then compute or give upper bounds for the linear complexity of several classes of graphs.
Combinatorial Interpretations Of Spanning Tree Identities, Arthur T. Benjamin, Carl R. Yerger
Combinatorial Interpretations Of Spanning Tree Identities, Arthur T. Benjamin, Carl R. Yerger
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We present a combinatorial proof that the wheel graph Wn has L2n − 2 spanning trees, where Ln is the nth Lucas number, and that the number of spanning trees of a related graph is a Fibonacci number. Our proofs avoid the use of induction, determinants, or the matrix tree theorem.
The Linking Probability Of Deep Spider-Web Networks, Nicholas Pippenger
The Linking Probability Of Deep Spider-Web Networks, Nicholas Pippenger
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We consider crossbar switching networks with base b (that is, constructed from b x b crossbar switches), scale k (that is, with bk inputs, bk outputs, and bk links between each consecutive pair of stages), and depth l (that is, with l stages). We assume that the crossbars are interconnected according to the spider-web pattern, whereby two diverging paths reconverge only after at least k stages. We assume that each vertex is independently idle with probability q, the vacancy probability. We assume that b ≥ 2 and the vacancy probability q are fixed, and that k …