All questions with tag [math: algebraic-graph-theory]


Properties of a generalized graph

I'll start with formulating my problem and then ask my question: To generalize a graph $Ga = (Va,Ea)$, we partition its nodes into disjoint sets. The elements of a partitioning $V$ are subsets of $Va$. They can be thought of as supernodes since they contain nodes from $Ga$, but are themselves the nodes of a undirected generalized graph $G = (V, ...
2022-07-21 09:03:35

Integrating matrix exponential

I have a question about equation 6 in this paper. Simplifying somewhat, the authors state the following $$\int_0^{\infty} e^{-tL} dt = L^{-1}$$ $L$ here is a graph laplacian and therefore is a matrix. (Leave aside for the moment that $L$ is singular and thus make the above equation meaningless.) To derive the equation, I first tried expanding $e...
2022-07-14 04:44:07

G graph of diameter d implies an adjacency matrix with at least d+1 distinct eigenvalues!

In reading the well known book on Algebraic graph theory I came across a theorem that could be stated in the following way: If $G$ is a graph of diameter $d$ then the adjacency matrix of $G$ has at least $d+1$ distinct eigenvalues. The proof shows that if $A$ is the adjacency matrix of a graph $G$ that has diameter $d$, then $I,A,A^2, \ldots, A^...
2022-07-12 11:36:54

Complexity of counting the number of triangles of a graph

The trivial approach of counting the number of triangles in a simple graph $G$ of order $n$ is to check for every triple $(x,y,z) \in {V(G)\choose 3}$ if $x,y,z$ forms a triangle. This procedure gives us the algorithmic complexity of $O(n^3)$. It is well known that if $A$ is the adjacency matrix of $G$ then the number of triangles in $G$ is $...
2022-07-11 23:39:37

wreath product and automorphism groups of graphs

I'm collaborating with automorphism teams of charts and also I have a trouble recognizing the wreath item of 2 teams. I additionally desire to see some instances of charts, apart from Kn, n, whose automorphism team include wreath item of Sn.
2022-07-09 23:35:07

Proof of Algebraic connectivity

I am really interested concerning the evidence of Algebraic connection Algebraic connectivity: The algebraic connection of a chart $G$ is the 2nd - tiniest eigenvalue of the Laplacian matrix of $G$ . This eigenvalue is more than $0$ if and also just if $G$ is a linked graph.This is an effect to the reality that the variety of times $0$ lo...
2022-07-01 16:01:02

"Semidirect product" of graphs?

The first subquestion is "has a standard notion of semidirect product been defined in graph theory"? If yes, i'd like to know if the definition i'm gonna give is equivalent to the standard one. I'd also like to know if there's some litterature about it. If the answer was "no", would you accept my definition as a reasonable &q...
2022-06-30 00:21:28

Graphs with eigenvalues of large multiplicity

For a highly normal chart, there are specifically 3 eigenvalues, all nonzero (I think). One has multiplicity 1, which suggests the various other 2 have rather high multiplicities. There are tables that offer these eigenvalues and also multiplicities: For instance, the Schlaefli chart is ord...
2022-06-28 23:06:39