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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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        23 views

        Distance between colored rooted graphs

        My question is what is the rough meaning of $\alpha_{1,2}$? I think of it in this way: Consider two balls one is around $o_1$ $B_{G_1}(o_1,[r])$ in the first graph and the other $B_{G_2}(o_2,[r])$ is ...
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        25 views

        Bayesian Networks and Polytree

        I am a bit puzzled by the use of polytree to infer a posterior in a Bayesian Network (BN). BN are defined as directed acyclic graphs. A polytree is DAG whose underlying undirected graph is a tree. ...
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        1answer
        131 views

        When is the poset of acyclic orientations of a graph a lattice?

        $\def\inv{\mathrm{inv}}\def\Acyc{\mathrm{Acyc}}$Let $G$ be a graph whose vertices are numbered $\{ 1,2, \ldots, n \}$. Given an orientation $\omega$ of $G$, define the inversions of $\omega$, written $...
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        1answer
        114 views

        Martingales and intersection of random walks

        Let $G=(V,E)$ be a graph with $n$ vertices. Consider a pair of independent simple random walks $(X,Y)$ on the graph, each of length $L$ starting from a node $v \in V$. We denote a length-$L$ random ...
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        23 views

        Beck-Fiala Discrepency Type Results for Arbitrary Graph Labelings

        Suppose we have a graph $G$ on $n$ vertices $x_1 , \dots, x_n$ attached with weights of values from $1$ to $n$. We will write $\text{weight}(x_i)$ as simply $x_i$ and let $\text{diff}(G) = \min _{(x_i,...
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        1answer
        80 views

        Elusive groups and vertex-transitive graphs

        This question is pertaining to finite connected vertex-transitive graphs. I recently read "Transitive permutation groups without semiregular subgroup" by Cameron, Giudici, Jones, Kantor, Klin, ...
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        47 views

        The graph polynomial of the Total Graph of a Graph

        Consider the Total Graph ($T(G)$) of a graph $G$ with vertex set $V$ edge set $E=\{(u,v)\}$ with Line graph $L(G)$ and subdivision graph $S(G)$ (formed by putting a vertex in each edge of the original ...
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        85 views

        Mod $2$ of $\#PM(G)$ for arbitrary G?

        Permanent mod $2$ of biadjacency gives polynomial time algorithm of $\#PM(G)\mod 2$ of perfect matchings of bipartite graph. Is there a similar efficient strategy for general graphs?
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        78 views

        Embedding a graph in $\mathbb{R}^3$ with partial geometric information

        I have a connected, sparse, graph (a molecule to be specific) and I'm interested in associating 3D coordinates with the vertices. Here's the kicker: I already have coordinates for none/some/all ...
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        1answer
        127 views

        Characterization of nilpotent adjacency matrices [closed]

        Let $\theta$ be the adjacency matrix of a simple graph (symmetric and zeros on the diagonal). What is the characterization of those $\theta$ which satisfy $$\theta^2 \equiv 0 \pmod{2}$$ i.e. which $\...
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        131 views

        Can the corollary of the Ihara–Bass formula be extended to $ u^2 = 1 $?

        Suppose there is a finite undirected graph $G(V,E)$ having $n$ vertices and $m$ edges. The non-backtracking matrix $B$ is indexed by $2m$ directed edges and defined as $$ B(a \to b, c \to d) = \delta_{...
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        1answer
        63 views

        Induced subgraphs of $\text{Exp}(G, K_2)$

        If $G, H$ are simple, undirected graphs, we define the exponential graph $\text{Exp}(G,H)$ to be the following graph: the vertex set is the set of all maps $f:V(G)\to V(H)$ two maps $f\neq g: V(G)\to ...
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        49 views

        Random graphs - multiple giant components

        For a given random graph, a connected component that contains a finite fraction of the entire graph’s vertices is called giant. A well known result in random graphs is the existence and uniqueness of ...
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        85 views

        Expander graphs with many 4-cycles

        The question is not strictly well-defined. But it goes like this: Could you find an infinite family of graphs $G_i$, that are all $\epsilon$-expanders, but have many 4-cycles? $\epsilon$ should ...
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        1answer
        122 views

        Do the Odd Cycles of a Graph Define a Matroid?

        An Odd Cycle Transversal is a set of vertices that, when removed from a graph, renders it bipartite. Question: does the collection of "critical" sets of vertices, whose removal renders a ...
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        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 ...
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        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 ...
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        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 ...
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        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 ...
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        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 ...
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        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 ...
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        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 $\...
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        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 $U^\top\sqrt{D^{-1}}(P+(\mathrm{e}^x-1)B(0)-\mu I)\sqrt{D}U$, ...
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        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 ...
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        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 ...
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        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....
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        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,...
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        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. ...
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        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 ...
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        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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        44 views

        Planar graphs with perfect matching count in linear time?

        We can find Pfaffian orientation and take determinant to compute permanent in $O(n^\omega)$ time where $\omega$ is exponent of matrix multiplication. We know that permanent of $O(n)$ vertex planar ...
        5
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        1answer
        117 views

        For what graph does the following algebraic property hold?

        Let $G=(V,E)$ be a simple graph. My question: For what graph $G$, does there exist a permutation $\sigma$ on $V$ such that $$\prod_{uv\in E}(x_{\sigma(u)}-x_{\sigma(v)})=-\prod_{uv\in E}(x_u-x_v)?$$ ...
        1
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        1answer
        100 views

        Bookthickness of covering space

        A book embedding of a graph G consists of placing the vertices of G on a spine and assigning edges of the graph to pages so that edges in the same page do not cross each other. The page number is a ...
        7
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        1answer
        141 views

        Co-spectral fractional isomorphic graphs with different Laplacian spectrum

        I am looking for two undirected graphs $G$ and $H$ of the same order (i.e., they have the same number of vertices) such that $G$ and $H$ are cospectral (i.e., their adjacency matrices $A_G$ and $A_H$ ...
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        1answer
        58 views

        A vertex transitive graph has a near perfect/ matching missing an independent set of vertices

        Consider a power of cycle graph $C_n^k\,\,,\frac{n}{2}>k\ge2$, represented as a Cayley graph with generating set $\{1,2,\ldots, k,n-k,\ldots,n-1\}$ on the Group $\mathbb{Z}_n$. Supposing I remove ...
        1
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        1answer
        69 views

        Shortest path on graphs

        I would like to now if there has been any work on related problems, that is, shortest path problem in dynamically evolving graphs.
        2
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        0answers
        80 views

        Can entropy of a network be written as a polynomial?

        In my research, I met a problem here. Consider a weighted graph Laplacian matrix $$\mathcal{L}_w(\mathcal{G}) = DWD^T,$$ where $D$ is a incidence matrix and $W$ is diagonal with each diagonal ...
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        0answers
        19 views

        Sufficient Condition for the Existence of Vertex Disjoint Shortest Paths

        Let $G(V,E)$ be a symmetric connected graph with $n$ vertices and let $D\in\mathbb{N}_0^{n\times n}$ be the matrix containing as entries $d_{uv}$ the least number of edges on a path from $u$ to $v$. ...
        1
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        1answer
        55 views

        Finding an element of the homology group of a graph which is everywhere nonzero

        Let $\Gamma$ be an oriented graph (multiple edges between vertices are allowed), and let $G$ be a finite abelian group. We define $H_1(\Gamma,G)$ to be the set of $G$-linear combinations of edges of $\...
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        1answer
        116 views

        Highly asymmetric regular graph

        Let $G$ be a regular connected simple graph on $n$ vertices with chromatic number $\chi$ and maximum degree $\Delta$. Then, it is implied that $G$ is $\chi$-partite. Suppose, we remove one of the ...
        5
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        1answer
        79 views

        Existence of regular factors in dense graphs

        All graphs here are finite and simple. A $d$-factor of a graph is a spanning regular subgraph of degree $d$. Where can I find theorems of this nature, for constants $a,b,c\gt 0$: If $G$ is a graph ...
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        49 views

        Finding fundamental Kempe-locking configurations

        Let $xy$ be an edge in a planar triangulation $T$ that is at least 4-connected and let $uxvy$ be the 4-cycle delineating the 4-face of the near-triangulation $G_{xy}$ obtained by deleting $xy$ from $T$...
        3
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        1answer
        67 views

        Equitable edge coloring of graphs

        Consider a simple regular graph $G$ with $n$ vertices and $E$ edges. Then, can we say that the edges can be colored equitably in $\Delta+1$ colors? By equitability is meant that a proper $\Delta+1$ ...
        5
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        0answers
        36 views

        Normal colorings of bridgeless cubic graphs

        Definition (informal) A normal edge-5-coloring of a bridgeless cubic graph $G$ is a proper 5 coloring of the edges of the graph, so that for each edge $e\in E(G)$, either $e$ and the four edges ...
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        51 views

        Helly vs Strong p-Helly Property of Hypergraphs

        I am not clear about the difference between Helly and Strong p-Helly property. For example hypergraph H(V, E), V = { 1,2,3 } and E = {(1,2), (2,3), (1,3)} has non-empty set for each pair of ...
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        58 views

        Bounds on chromatic number when maximum degree is large

        For a regular graph with $n$ vertices and maximum degree $\Delta$, it is easy to see that the chromatic number, $\chi\le\frac{n}{2}$ if $\frac{n}{2}\le\Delta\lt n-1$(since a regular graph on $n$ ...
        2
        votes
        1answer
        115 views

        Chromatic number of the linear graph on $[\omega]^\omega$

        Let $[\omega]^\omega$ denote the set of infinite subsets of $\omega$. Let $$E = \{\{a,b\}: a,b\in [\omega]^\omega\text{ and } |a\cap b| = 1\}.$$ It is clear that $G = ([\omega]^\omega, E)$ has no ...
        1
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        1answer
        39 views

        Clarifications regarding conformability in graph colorings

        As an outgrowth of this question, I have another question, that is, why not the definition of conformability includes a $\Delta$ vertex coloring also, instead of only $\Delta+1$ coloring of vertices. ...
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        78 views

        A simple case of a strong version of the Berge-Fulkerson conjecture

        UPDATE 28 June 2019 A counterexample for Conjecture 2 has been provided. The conjecture is now demoted again to guess. The text has been updated to reflect this change, and there is now a new ...
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        19 views

        Perfect matchings and edge cuts in cubic graphs - part 1

        Let $G$ be a bridgeless cubic (simple) graph, and let $M$ be a perfect matching in $G$. $G-M$ will necessarily be a set of circuits. For example, if we delete a perfect matching from $K_{3,3}$ we ...

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