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Questions about properties of and problems on graphs, discrete data structures that have the form of nodes connected by edges, that is networks.

3
votes
Let $d_G(x,y)$ be the distance between the vertices $x,y \in V(G)$ for a connected graph $G.$ The characteristic path length is then defined as $$\frac{\sum_{x,y \in V(G)} d_G(x,y)}{n(n-1)} = \frac{\s …
answered Dec 21 '12 by Jernej
18
votes
I am not sure what bothers you but as I see it you are confused about the following two facts If a graph is connected then $e \geq n-1.$ If a graph has more than $e > \frac{(n-1)(n-2)}{2}$ then it …
answered Nov 21 '12 by Jernej
17
votes
Isomorphism formalizes the notion of equal graphs. For example on this figure you see three isomorphic graphs More formally, an isomorphism of graphs $G_1$ and $G_2$ is a bijection $f:V(G_1) \mapsto …
answered Jan 2 '13 by Jernej
7
votes
The best way (in terms of laziness) is to use the freely available tool Sage which has the best support for graph theory. Example sage: G = graphs.PetersenGraph() sage: G.has_homomorphism_to(graphs …
answered Oct 25 '13 by Jernej
5
votes
The answer is no. A counterexample is given by the following two graphs both having $(-1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 …
answered Sep 5 '13 by Jernej
10
votes
Suppose you wish to compute the random graph $G(n,p)$ that is the graph with $n$ vertices where each edge is added with probability $p.$ Suppose you have a coin that gives tails with probability $p$ …
answered Dec 28 '12 by Jernej
6
votes
Let $R(s,t)$ be the least integer $k$ such that every graph on $k$ or more vertices contains either a $s$-clique or independent set of size $t.$ It turns out that this number is well defined (called …
answered Aug 25 '13 by Jernej
5
votes
There are no general algorithms for your problem but what one usually does (for graphs of small orders) is Use nauty to generate graphs satisfying some rough constraints (you can make nauty generat …
answered Nov 21 '12 by Jernej
7
votes
The short answer is no. No quick algorithm for this problem is known. A big open problem (for at least 50 years) in algebraic graph theory asks about the existence a regular graph of degree 57 orde …
answered Oct 25 '12 by Jernej
4
votes
Your thinking is correct. To verify your claim you can check this lecture notes on the probabilistic method by Matousek and Vondrak. More specifically check the proof of Theorem 3.3.1 stating that any …
answered Dec 16 '12 by Jernej