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Alternating cycles in edge-partitioned graphs
TLDR
We show that if edges of a 2-connected graph G are partitioned into two classes so that every vertex is incident with edges from both classes, then G has an alternating cycle. Expand
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A portion of the well-known collaboration graph
TLDR
The collaboration graph C has as vertices all researchers (mathematicians, in particular), with an edge joining every pair of authors of a multi-author paper if u and v have published a joint research paper. Expand
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Famous trails to Paul Erdős
TLDR
The notion of Erd} os number has oated around the mathematical research community for more than thirty years, as a way to quantify the common knowledge that mathematical and scientiic research has become a very collaborative process in the twentieth century, not an activity engaged in solely by isolated individuals. Expand
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On weakly connected domination in graphs
TLDR
A dominating set D is a weakly connected dominating set of a connected graph G if (V,E@?(DxV)) is connected. Expand
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The evolution of the mathematical research collaboration graph
TLDR
We discuss some properties of the research collaboration graph for mathematicians, look at its evolution over time, and survey some random models that might produce graphs of this sort. Expand
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Small Worlds: The Dynamics of Networks between Order and Randomness. By Duncan J. Watts
TLDR
Small Worlds: The Dynamics of Networks between Order and Randomness. Expand
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Generalized matrix tree theorem for mixed graphs
In this article we provide a combinatorial description of an arbitrary minor of the Laplacian matrix (L) of a mixed graph (a graph with some oriented and some unoriented edges). This is a generalizedExpand
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On the minors of an incidence matrix and its Smith normal form
Abstract Consider the vertex-edge incidence matrix of an arbitrary undirected, loopless graph. We completely determine the possible minors of such a matrix. These depend on the maximum number ofExpand
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Algebraic graph theory without orientation
Abstract Let G be an undirected graph with vertices {v 1 ,v 2 ,…,>;v ⋎ } and edges {e1,e2, …,eϵ}. Let M be the ⋎ × ϵ matrix whose ijth entry is 1 if ej is a link incident with vi, 2 if ej is a loopExpand
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Fractional Arboricity, Strength, and Principal Partitions in Graphs and Matroids
TLDR
We characterize the graphs and matroids G for which γ(G) = η(G). Expand
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