I am going through Quantum factoring, discrete logarithms and the hidden subgroup problem by Richard Jozsa. On page 13, the author discussed the hidden subgroup problem (HSP) formulation of the graph isomorphism (GI) problem. I would like to make it sure that I get the development of the concept right. Here both $A$ and $B$ […]

Consider the following problem Let $n \leq 5$ and let $\Gamma = \mathrm{Cay}(C_{2n},S)$ be the Cayley graph with Cayley set $S$. Show that $\Gamma$ is isomorphic to $\mathrm{Cay}(D_{2n},S’)$ for a suitable $S’.$ Recall that $\Gamma = \mathrm{Cay}(G,S)$ is a Cayley graph with vertex set $G$ if $G$ is a group, $S$ is a subset of […]

suppose $n$ people are in a party and every two of them have exactly one common friends,prove that there is one who is friend to all. I suppose there is no one who is friend to all,I want to show that the graph with this assumption is strongly regular graph and then show that two […]

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 […]

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 […]

I am reading “Graphs of Degree Three with given Abstract Group” (by Robert Frucht) where the author describes (somewhat tedious) algorithms to construct suitable graphs starting from a given group. I would like to see a graph associated with the quaternion group as its automorphism group. Following the given algorithm the graph should have 32 […]

We are given a finite group $G$ and wish to find a DAG (directed acyclic graph) $(V,E)$ whose automorphism group is exactly G (a graph automorphism of a graph is a bijective function $f:V\to V$ such that $(u,v)\in E \iff (f(u),f(v))\in E$). A similar (positive) result for undirected graph is known: Frucht’s theorem. My uneducated […]

I am reading “Algebraic Graph Theory” by Norman Biggs (1974). On page 119, there is a proposition which says the following: Proposition 18.1: Let $[\alpha]$ be a $t$-arc in a cubic $t$-transitive graph $X$. Then an automorphism of $X$ which fixes $[\alpha]$ must be the identity automorphism. To understand the Proposition, and the proof, some […]

Can anybody help me in clearing the doubt about hierarchical product of graphs. Its quite different from other graph products. Here is the screenshot and link how it is done. I know the rooted graph, but its applied here by using 0 in definition and in adjacency relation is out of my mind. I will […]

I had the need to calculate the adjacency matrix $L$ of the line graph of a certain planar $k$-regular graphs $G(n,e)$ ( $n$ vertices and $e=\frac k2 n$ edges) given its adjacency matrix $A_G$. Here I went: Let $N_k$ an “outer-diagonal” indexing matrix, e.g. $N_4=\pmatrix{0&1&2&3\\1&0&4&5\\2&4&0&6\\3&5&6&0}$. Calculate the Hadamard Dot product $B=N_k\odot A$. I found that […]

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