Two graphs $G$ and $H$ are isomorphic if they have a function $f$ which provides an exact pairing of vertices in the two graphs such that for any adjacent vertices $u,v\in \{\mbox{set of vertices of }G\}$, $f(u)$ and $f(v)$ are also adjacent in $H$.

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Graph Isomorphism of Complete Graph.

what Is the complexity(computational complexity) of graph isomorphism of 1.Complete graphs($K_n$) and 2.Utility graphs (Complete bipartite graphs ,$K_{n,n}$)? is it in polynomial ? Looks ...
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91 views

Simple connected plane graph G and its dual graph G*; if G is isomorphic to G*, then G is not bipartite?

Let $G$ be a simple connected plane graph where $v>2$, and $G^*$ is its dual graph. Prove that if $G$ is isomorphic to $G^*$, then $G$ is not bipartite. I know that $G$'s number of faces is ...
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Reorder adjacency matrices of regular graphs so they are the same

Given a matrix A of a strongly $k$ regular graph G(srg($n,k,\lambda,\mu$);$\lambda ,\mu >0;k>3$). The matrix A can be divided into 4 sub matrices based on adjacency of vertex $x \in G$. $A_x$ ...
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55 views

Do cycle graphs determine groups up to isomorphism?

This erroneous version of the wikipedia article for Cycle Graphs stated that the cycle graph of different groups could be the same. The immediate error is that it stated that the cycle graph of ...
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1answer
22 views

Draw all non-isomorph graphs with 7 knots

in our course we had to draw all non-isomorph graphs with 7 knots without loops. Since this is hard to explain I'll add an Image. The blue written part is accepted by the tutor and the pencil ...
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49 views

How to determine if these graphs are isomorphic?

I had this question on my last Discrete exam: (the missing vertex on graph G is vertex d) I did prove that the graphs were isomorphic, but my teacher said that I matched up my vertices wrong. ...
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0answers
32 views

Nauty graph isomorphism weighted edges

I want to use sofeware Nauty version 2.5 to check if edge-weighted graphs are isomorphic to each other. However weighted edge is not supported by Nauty. Professor McKay mentioned that there's an ...
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1answer
41 views

What is Graph Isomorphism and Graph Invariant?

While I was reading Reinhard Diestel text on graphs, I came across this paragraph. Let G = (V, E) and G' = (V' , E' ) be two graphs. We call G and G' isomorphic,and write G $\simeq $ G' , if ...
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1answer
65 views

All non-isomorphic graphs

How can I find all non-isomorphic graphs with 2, 3 or 4 nodes, including not-connected ones? I know that to graphs $G_1$ and $G_2$ are isomorphic, if there is a bijective depiction $f: V(G_1) ...
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0answers
22 views

Is there a polynomial time algorithm for Poly-trees (oriented trees) isomorphism?

In terms of graph isomorphism complexity classes Trees have a polynomial time algorithm and Directed Acyclic Graphs (DAG's) do not (so far). What about Poly-trees (oriented trees)? These are DAG's ...
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1answer
17 views

Subgraph isomorphism problem

Subgraph isomorphism problem is an NP-hard problem. However, if the subgraph size is constant (assume $k$), then it can be polynomial time solvable. The most easiest way is that: Randomly obtain $k$ ...
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2answers
343 views

What's the difference between the automorphism and isomorphism of graph?

What's the difference between the automorphism and isomorphism of graph? In graph theory, an isomorphism of graphs $G$ and $H$ is a bijection between the vertex sets of $G$ and $H$, $ f \colon ...
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1answer
37 views

How to find that two adjacency matrices are equal

What is the easiest way to tell if these two graphs are isomorphic and how do I know which nodes in both graphs are the same. I've made the adjacency matrices but they are pretty big. I think I need ...
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4answers
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Prove two graphs are isomorphic

I have identified two ways of showing it isomorphic but since it is a 9 mark question I dont think i have enough and neither has our teacher explained or given us enough notes on how it can be ...
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2answers
46 views

Why are these two graphs isomorphic?

I have the following graphs and I am trying to identify if they are isomorphic or not. The solutions that I have indicate they are isomorphic, with the following bijection: However, it appears ...
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1answer
639 views

How can I tell how many non-isomorphic unrooted trees with 6 edges exists without drawing them all?

Typically my professor asks that we draw them all, but I would like to save some time to confirm how many I need.
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3answers
71 views

Nice Isomorphic Question and way of finding Quickly?

I see a question about Isomorphic Graph. the question is: The first line graph is not isomorphic with which one in the second line? How we can find isomorphic graph without knowing some geometry map ...
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1answer
49 views

Creating a Bijection to check if Graphs are Isomorphic

To prove that two graphs are isomorphic I was taught to first consider the bijection between the two graphs. I was never taught however the rules when coming up with the bijection. Is my only rule, ...
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0answers
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are all transitive reductions of a graph isomorphic?

Transitive reduction of a graph is not unique. Take this example: Now if we remove the labels, the two reduced graphs are isomorphic: are all transitive reductions of a directed graph isomorphic ...
2
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0answers
46 views

Why is the graph of 4 nodes and 2 edges not self-complementary?

I am having some trouble seeing why a graph of 4 nodes and 2 edges is not self-complementary such that G is isomorphic to G bar (G complement) (please see the attachment below). I know that the number ...
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1answer
27 views

Reducing Subgraph Isomorphism Complexity via an Enforced Layout of Nodes

This is my first question on the mathematics stack exchange. I hope my question is at least a little interesting, then. In any case, I was reading up on the subgraph isomorphism problem on wikipedia. ...
2
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1answer
39 views

Adding an edge and a vertex to non-isomorphic graphs

Let $G$ and $H$ be two non-isomorphic simple graphs of equal order and equal size. Suppose I am to add a vertex $v$ and and edge $e$ incident to $v$ to $G$ and $H$. By add I mean to connect $v$ to ...
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1answer
50 views

What is a graph isomorphism?

I am trying to under isomorphism in graphs, and from what I know, if graph A is isomorphic to graph B, then you could basically just rearrange the nodes in A, while keeping the edges connected the ...
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0answers
25 views

Isomorphism testing for minimal SP-trees

I'm doing a bit of research on the SP-trees. I'm still new to this whole problematic, so I'd be thankful if someone cleared this thing up. :) This is the scenario that I'm trying to come up with a ...
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3answers
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Graph isomorphism when all vertices have the same degree

Are 2 connected graphs isomorphic if they have the same number of vertices and each vertex has the same degree $k$? I don't know how to prove it but I also can't find a counter example.
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1answer
94 views

How many nonisomorphic graphs are there with 10 vertices and 43 edges?

How would I go about solving this? I know that $K_{10}$ has $9+8+7+\dots+1=45$ edges. So would it be something like $\binom {45}{43}$ because out of the 45 total edges, one must choose 43 for the ...
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4answers
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Are these 2 graphs isomorphic?

They meet the requirements of both having an $=$ number of vertices ($7$). They both have the same number of edges ($9$). They both have $3$ vertices of degree $2$ and $4$ of degree $3$. However, ...
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1answer
63 views

Isomorphisms of infinite graphs that are fixed on a set of vertices

Suppose we have two infinite directed graphs $\langle V,E\rangle$ and $\langle V,E^*\rangle$, and a set $A \subseteq V$ such that, for all finite $X \subseteq A$, there is an isomorphism from $\langle ...
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44 views
3
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1answer
3k views

How to draw all nonisomorphic trees with n vertices?

I have a textbook solution with little to no explanation (this is with n = 5): Could anyone explain how to "think" when solving this kind of a problem? (for example, drawing all non isomorphic ...
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votes
0answers
75 views

Show that the number of isomorphism classes of tree on n vertices is exponentially large as a function on n.

I am currently trying to answer this question: 'Show that the number of isomorphism classes of tree on n vertices is at least $\frac{n^{n−2}}{n!}$ and hence that this is exponentially large as a ...
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1answer
249 views

Number of isomorphism classes of a tree on n vertices

I'm currently trying to solve this problem: "Show that the number of isomorphism classes of tree on n vertices is at least $\frac{n^{n-2}}{n!}$." I'm pretty stumped to be honest. I know of Cayley's ...
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1answer
46 views

Vertex degree and graph isomorphism

Suppose I have two simple graphs $G(V,E)$ and $H(V,E)$ with number of vertices $N$. And $\forall i \quad \text{such that}\quad 0<i<N$ No:of elements in $V(G)$ with degree $i $ = No:of elements ...
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2answers
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Concept: The graph component

I have the following definition for a Component of a graph: A subgraph $H$ of a graph $G$ is a component of $G$ if $H$ is a maximal connected subgraph of $G$, ...
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1answer
64 views

Is the number of simple circuits of a particular length preserved in two isomorphic graphs?

If two graphs are isomorphic, and one has a simple circuit of a particular length, must the other graph also have a circuit of the same length? Do they also have to have the same number of such ...
0
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1answer
70 views

Any two abelian group of order 8 must be isomorphic

TRUE/FALSE :Any two abelian group of order 8 must be isomorphic SOLUTION: True The problem of finding all abelian groups of order 8 is impossible to solve, because there are infinitely many ...
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1answer
68 views

Proving two graphs are isomorphic in polynomial time - Bondy/Murty - Graph Theory Page 6

I am trying to do the below problem: Now I can't see how one does this. I know you can explicitly show the bijections, but I can't see an easy way to do this, since it is $3\text{-regular}$. I ...
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1answer
60 views

Proof verification: (Pretty pictures :) )Showing two graphs are not isomorphic. Bondy/Murty - Graph theory Page 5

I want to show the below two graphs are not isomorphic. Treat the left as graph $\bf G$ and the right as graph $\bf H$. $\bf G$ and $\bf H$ are not isomorhpic: Although we have a bijective mapping ...
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2answers
108 views

How to calculate the number of automorphisms of a given graph?

How do determine the number of isomorphisms that a graph has to itself? For instance, suppose we have the following graph: How do I determine how many isomorphisms there are from G itself?
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1answer
52 views

$3$-connected graph isomorphism problem

If $G$ is $3$-connected, then $G$ contains a subgraph isomorphic to $H$, where $H$ is obtained from $K_4$ by replacing the edges of $K_4$ by internally disjoint paths. Any hints and proofs are ...
0
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1answer
39 views

How can I show complete graphs are determined by spectrum?

I understand how to prove a complete graph $K_n$ has spectrum $\lbrace -1^{(n-1)},n-1 \rbrace$. However I am having difficulty proving that the spectrum uniquely determines the complete graph. ...
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2answers
52 views

Are these graphs isomorphic?

Are these 2 graphs isomorphic? (sorry for the bad picture quality) For the solution: 1) They both have 8 vertices 2) They both have 12 edges 3) They both have 8 vertices of degree 3 4) Is this ...
1
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1answer
23 views

Nonisomorphic graphs Discrete Math

Find 3 different nonisomorphic graphs with 8 vertices that have the degree sequence 2,2,2,2,2,2,2,2. Answer: An 8-cycle, or two 4-cycles, or a 3-cycle and a 5-cycle. Can someone show me how these ...
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1answer
25 views

isomorphism of graph

I have a doubt about an example of isomorphism of graphs. I am looking at the notes of group theory by Donald Kreher. (http://www.math.mtu.edu/~kreher/ABOUTME/syllabus/GTN.pdf). Please see page 9 ...
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2answers
143 views

Isomorphism vs equality of graphs

I have just started studying graph theory and having trouble with understanding the difference b/w isomorphism and equality of two graphs.According what I have studied so far, I am able to conclude ...
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0answers
38 views

How many different graphs are there with a given degree-sequence?

Let G be a simple undirected graph. If it has at most $4$ edges, there is only one graph for any degree-sequence, but for $n = 5$, the situation changes. There are $34$ different (non-isomorphic) ...
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1answer
46 views

A graph with minimum degree $k+2$ contains any $(k+3)$-vertex tree as a subgraph?

Let $k$ be a positive integer and let $T$ be a tree of order $k+3$. If $G$ is a graph with minimum degree at least $k+2$, prove that $G$ contains a subgraph isomorphic to $T$. Any solutions or hints ...
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1answer
137 views

Calculating the number of isomorphic classes of complete bipartite graph

How many isomorphism classes of complete bipartite graphs have exactly 10 vertices? I don't understand what the question is asking or how to go about solving it.
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1answer
65 views

Graph isomorphism problem for labeled graphs

In the case of unlabeled graphs, the graph isomorphism problem can be tackled by a number of algorithms which perform very well in practice. That is, although the worst case running time is ...