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        Questions tagged [graph-theory]

        Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, topological-graph-theory, random-graphs, graph-colorings and several others.

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        4
        votes
        1answer
        179 views

        The list chromatic number of some special toroidal grid graphs

        A list-assignment $L$ to the vertices of $G$ is the assignment of a list set $L(v)$ of colours to every vertex $v$ of $G$; and a $k$-list-assignment is a list-assignment such that $|L(v)|\geq k$, for ...
        -1
        votes
        0answers
        28 views

        Correlated tree interval and existence of unary subtree

        We have a collection of random intervals $\{I_{k}:=(X_{k},Y_{k})\}_{k=1}^{\infty}\subset [0,1]$ s.t. For deterministic $l_{k}\to 0$ we have $0<l_{k}^{a_{1}}\leq Y_{k}-X_{k}\leq l_{k}^{a_{2}}$. The ...
        2
        votes
        2answers
        119 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 ...
        7
        votes
        0answers
        135 views

        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 ...
        3
        votes
        0answers
        48 views

        Polynomial Graph Isomorphism from Star System Reconstruction?

        Confusion is possible, but two papers and a simple graph transformation imply Graph Isomorphism is polynomial, which is an open problem. The closed neighborhood of a vertex in a graph is sometimes ...
        1
        vote
        0answers
        37 views

        A regular independence induced graph in a $\Delta+1$ coloring

        Consider any regular graph $G$ with order $n$ and size $E$ and maximum degree $\Delta$. Now, we give a $\Delta+1$ coloring to the vertices such that each vertex and its neighbors receive distinct ...
        3
        votes
        1answer
        226 views

        Asymptotic formula for the number of connected graphs

        It can be shown that the set of graphs with $N$ vertices $G_N$ has cardinality: \begin{equation} \lvert G_N \rvert = 2^{N \choose 2} \tag{1} \end{equation} Recently, I wondered how much bigger $\...
        7
        votes
        1answer
        327 views

        Understanding Gillman's proof of the Chernoff bound for expander graphs

        My question is about the proof of Claim 1 in this paper: Gillman (1993). At the end of the proof, the author says: The matrix product $U^\top\sqrt{D^{-1}}(P+(\mathrm{e}^x-1)B(0)-\mu I)\sqrt{D}U$, ...
        2
        votes
        1answer
        129 views

        Proving a theorem on coloring a peculiar graph

        Consider the graph formed by $k$ cliques of order $k$, any two cliques sharing at most one point in common. Now, by Szekeres-Wilf theorem, I think the graph should be $k$ colorable, as any connected ...
        4
        votes
        1answer
        141 views

        Hamiltonian paths on the space of graphs

        Disclaimer: I am not a professional graph theorist. Motivation: Let's consider the set $G_N$ of graphs with $N$ vertices where the vertices are assumed to be distinguishable. This set may ...
        0
        votes
        1answer
        62 views

        Latent Dirichlet allocation and properties of digamma function

        In the paper Blei, D. M., Ng, A. Y., & Jordan, M. I. (2003). Latent Dirichlet Allocation. Journal of Machine Learning Research, 3(4–5), 993–1022. http://www.jmlr.org/papers/volume3/blei03a/blei03a....
        12
        votes
        0answers
        229 views

        Is there an algorithm to compute a Belyi map for the Riemann surface?

        Let $y^2=x^5-x-1$ be an affine model of a projective complex curve, is there an algorithm to compute the Belyi map (preferably of small degree), i.e., map to the projective line ramified only at $\{0,...
        3
        votes
        1answer
        78 views

        Strong chromatic index of some cubic graphs

        Edit 2019 June 26 New computer evidence forces us to revise our guesses relating strong chromatic index and girth Edit 2019 June 25 Some mistakes have been corrected. Question 2 has changed. ...
        2
        votes
        0answers
        45 views

        Calculating Minimum Spanning Trees in Very Big Graphs

        I need to determine Minimum Spanning Trees (MST) of very big complete graphs, whose edgeweights can be calculated from data that is associated with the vertices. In the planar euclidean case, for ...
        2
        votes
        0answers
        17 views

        Complexity of weighted fractional edge coloring

        Given an edge-weighted multigraph $G=(V,E)$ with a positive, rational weight function $(w(e): e \in E)$, the weighted fractional edge coloring problem (WFECP) is to compute ($\min 1^T x$ subject to $...

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