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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

        Graph topology of random and ordered sphere packing

        Context: Sphere packing problems are often formulated in terms of optimization problems, where for a given dimensionality, one is interested in finding out the densest possible packing of the spheres ...
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        1answer
        44 views

        Sum-balanceable finite graphs

        Let $G=(V,E)$ be a finite simple, undirected graph. Given $f:V\to \mathbb{Z}$ we define the neighborhood sum function $\mathrm{nsum}_f:V\to\mathbb{Z}$ by setting $\mathrm{nsum}_f(v) = \sum\{f(w):w\...
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        19 views

        questions about message passing derivation from factor graph

        For the Machine Learning book, https://www.microsoft.com/en-us/research/uploads/prod/2006/01/Bishop-Pattern-Recognition-and-Machine-Learning-2006.pdf, On page 405 about the message passing algorithm ...
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        35 views

        3-edge colorings for special cubic graphs on a double torus

        In Complexity of 3-edge-coloring in the class of cubic graphs with a polyhedral embedding in an orientable surface it is proven that the task to find these colorings is NP-complete in the general case,...
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        32 views

        Behaviour of global clustering for common random graph models

        In order to develop some intuition for some of the commonly used random graph models, I've been looking at the global clustering coefficient as a means of comparing them. In particular, for the ...
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        74 views

        The properties of almost all directed graphs

        A mathematician on the forum previously requested a reference on human brains modelled as directed graphs. This makes sense as neurons are mostly unidirectional and I have been thinking about similar ...
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        1answer
        121 views

        Chromatic number and graph polynomial

        If $\prod_{i=1}^t x_i^{e_i}$ is a monomial, define $$rad\biggl(\prod_{i=1}^t x_i^{e_i}\biggr)$$ to be the number of distinct (nonzero) values of $e_i$. Now let $G$ be a simple graph with vertices ...
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        1answer
        92 views

        Almost all simple graphs are small world networks

        Two days ago it occurred to me that almost all simple graphs are small world networks in the sense that if $G_N$ is a simple graph with $N$ nodes sampled from the Erd?s-Rényi random graph distribution ...
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        72 views

        Strong tournaments

        Let $T$ be a strong tournament, and let $E=v_1v_2 \cdots v_n$ be an enumeration of $V(T)$. Let $C$ be a circuit in $T$. We define $i_E(C)=|\{(v_i,v_j) \in E(C); i>j\}|$. Suppose that $E$ is chosen ...
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        34 views

        A weakened form of list coloring

        The list coloring of a simple loopless graph is the assigning of a color from a certain list of colors to every vertex. The list coloring chromatic number of a graph is the minimal cardinality of the ...
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        1answer
        40 views

        Number of maximal independent sets in a simple graph

        Consider a simple regular graph on $n$ vertices and size $E$. How many distinct maximal independents can we find at the least in the graph? I think we can always find at least two maximal ...
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        1answer
        82 views

        How many nodes does a ball of radius $r$ in the Johnson graph $J(n,k)$ contain?

        1) How many nodes does a ball of radius $r$ in the Johnson graph $J(n,k)$ contain (Volume)? 2) How many nodes $v$ does a ball with center $x$ of radius $r$ in the Johnson graph $J(n,k)$ contain such ...
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        1answer
        64 views

        Energy of a symmetric matrix with $0$, $1$ or $-1$ entries

        I have a symmetric matrix with entries $0$, $1$ or $-1$ which appeared in my works in graph theory (the diagonal entries are all zero). I need a good upper bound for the energy of this matrix; i.e. "...
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        156 views

        Cardinality of a set of mutually disjoint perfect matchings of $K_\omega$

        If $G=(V,E)$ is a simple, undirected graph, we say that $M\subseteq E$ is a perfect matching if the members of $M$ are pairwise disjoint and $\bigcup M = V$. Let $K_\omega$ be the complete graph on $\...
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        58 views

        Faithful Orthogonality Dimension of Kneser Graphs

        Let us consider the complement of the Kneser graph with parameters $n$ and $n/4$. The vertex set of our graph $K$ is the set $\binom{[n]}{n/4}$ of $n/4$-subsets of $[n]$, and two vertices are joined ...

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