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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66 views

How to determine number of isomorphic classes of simple graph with n vertices, each with degree m?

For HW, I need to find the number of isomorphic classes of a simple graph with 7 vertices, each with degree two. I know I could brute-force it by finding all edge sets that fulfill that criteria, but ...
2
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1answer
294 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 ...
-1
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1answer
15 views

Given three non-isomorphic spanning trees of the complete graph K5, how many trees in each class?

The graph, K5 has 125 different spanning trees, which I beleive fit into three different non-isomorphic classes of spanning trees. However, I'm at odds as to how to figure out how many are in each ...
0
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3answers
51 views

Show that the polynomial ring in $n-1$ variables is isomorphic to the polynomial ring in one variable $x_{n}$

For a ring $R$ and for $n \geq 1$, define $ S := R[x_{1},...,x_{n-1}]$ for the polynomial ring in $n-1$ variables with coefficients in $R$. Show that $R[x_{1},...,x_{n}]$ is isomorphic to the ...
0
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1answer
28 views

Number of non isomorphic graphs

Let $S$ be a set of all graphs with 30 vertices and 432 edges. Find how many of them are mutually non isomorphic. My approach: we can look at their complements instead. Since $K_{30}$ has ...
1
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1answer
47 views

Hypercubes and bipartite graphs not isomorphic to subgraph of k-cube

Is hypercube and k-cube the same? I did see the question in another post here, but I am not able to comment there since I do not have much reputation, and I am not allowed to post a doubt as answer. ...
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2answers
35 views

Are these two graphs isomorphic?

I have the attached the images of two graphs. I want to know whether two graphs are planar or not. ? I also want to know whether two graphs are planar or not ?
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3answers
65 views

Find an isomorphism between two graphs

Let $G_1$, $G_2$ be the graphs shown below: Decide if $G_1$ and $G_2$ are isomorphic. If so, exhibit an isomorphism. Otherwise exhibit an invariant property for isomorphism that one of the ...
2
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2answers
35 views

Prove or disprove: involves Chromatic numbers, and subgraphs isomorphic to Kr

Prove or Disprove, a) if a graph $G$ contains a subgraph isomorphic to $K_r$, then the chromatic number is greater than or equal to $r$ b) if the chromatic number is great than or equal to $r$, then ...
0
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1answer
26 views

Proving the number of subgraphs of $G$ isomorphic to $F$

Let $F$ and $G$ be graphs. Let $sub(F, G)$ denotes the number of subgraphs of $G$ that are isomorphic to $F$, let $inj(F, G)$ denote the number of injective homomorphisms from $F$ to $G$ and let ...
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1answer
53 views

Give reason why a graph is not isomorphic [closed]

I need to give an example of two connected graphs with the same degree that arent isomorphic and say why they are not isomorphic
3
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2answers
67 views

Is $\exp:\mathbb{M_n}\to\mathbb{M_n}$ injective?

This is related to a personal exploration of isometries of directed graphs, motivated by my son's Lego Duplo train tracks and identifying "interesting" layouts. If $M$ is the adjacency matrix for a ...
0
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0answers
43 views

Given a graph, is there a way to find the number of isomorphisms?

As the question asks, is there a known way to find the number of isomorphisms of a given graph? I can't find a link anywhere (although I can find the number of distinct isomorphism classes for a graph ...
0
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0answers
52 views

Why is a connected planar graph isomorphic to its double dual?

Let $G^*$ be the dual graph of a planar graph $G$ (see wikipedia article). How does one prove that if $G$ is connected then it is isomorphic to $G^{**}$?
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1answer
76 views

Graph Theory: Isomorphic graphs

Show that the inverse of an isomorphism of graphs is also an isomorphism of graphs. So I just started a graph theory course and am having a little trouble with one of the problems on the homework. I ...
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1answer
56 views

Graph isomorphism

The definition for graph isomorphism states that two graphs $G_1 = (V_1,E_1)$, $G_2 = (V_2,E_2)$ are isomorphic if there is an isomorphism $f:V_1\to V_2$ such as each $u$ and $v$ vertex from $V_1$ are ...
1
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2answers
63 views

Determining if two graphs are isomorphic

I'm supposed to determine if the above graphs are isomorphic. I thought there was because there was a bijection from the set of vertices of graph G to the set of vertices of graph H, and because ...
0
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1answer
31 views

Isomorphism of Complete Graphs

I am struggling to understand the concept of isomorphism. By definition, if G and H are two simple graphs so that V(g) and V(h) are the number of nodes in G and H respectively, then isomorphism is ...
0
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1answer
46 views

How do I prove isomorphism?

I need to prove this: $$S_{\mathbb{N}}\cong S_{\mathbb{Z}}$$ ($S$ means permutation). I'd like to get ideas how to prove it... Thank you!
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2answers
48 views

Restriction on Graph Automorphism

This question referes to a definition in Eugene M. Luks paper "Isomorphism of Graphs of Bounded Valence Can Be Tested in Polynomial Time" (1981), page 48, available at ...
0
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0answers
34 views

proving that a given graph is isomorphic to an arbitrary graph with same satisfied condtions

Prove that any graph with satisfies the following conditions: each vertex has degree 3 any two adjacent vertices do not have common neighbors ( so no 3-cycles) any two non-adjacent vertices share ...
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votes
2answers
50 views

Which of those are isomorphism [closed]

I have here a list. I need to prove which one of them is true (or not) and prove it... $\mathbb{Z}_{21}\times \mathbb{Z}_{50}^*\cong\mathbb{Z}_{420}$ ...
1
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2answers
63 views

Are these graphs Isomorphic

please consider this two graphes. G1: G2: Are they Isomorphic? Is G1 a planer graph? It contains a K 3,3 or k5? thanks alot
0
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3answers
98 views

Show two graphs are not isomorphic

I know this graphs are not isomorphic. However they have the same number of vertex and edges, and the same degree sequence, is not the most easy case. If im correct, the graphs are isomorphic if ...
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1answer
53 views

Proof on ring isomorphism- irreducible

Consider the ring isomorphism $\phi: A \to B$ I have to prove that $a\in A $ is irreducible if and only if $\phi(a)$ is irreducible. By definition, $a$ is irreducible in A if and only if: 1) $a$ ...
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1answer
59 views

Graph of unbounded degree?

I was reading about The Graph Isomorphism Problem on Wikipedia and the article lists a number of special cases for which the problem can be solved in polynomial time. One of these cases is a graph of ...
0
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1answer
44 views

Non-Isomorph trees of a graph

Please consider this graph How many non-Isomorph trees with 4 vertex has this graph? Is there any formula that show number of non-Isomorph trees with $n$ vertices? thanks
1
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1answer
71 views

Non-isomorphic simple graphs: order $n$, size $\displaystyle \frac{na}{2}$, degree sequence $(a,a,a,…,a) \in \mathbb{N}^n$

If a simple graph has order $n$, size $\displaystyle \frac{na}{2}$ and degree sequence $(a,a,a,...,a) \in \mathbb{N}^n$ then is it unique up to isomorphism? I thought of this question while ...
2
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2answers
27 views

What are multiple isomorphisms?

For example, this graph has "multiple isomorphisms." What does that mean? And could you list them? I don't understand how there can be more than one.
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2answers
95 views

Prove that the group G defined by a~b=a+b+ab is isomorphic to the multiplicative group of nonzero real numbers.

Question: Prove that the group $G$ consisting of the set $\mathbb{R}\setminus\{-1\}$ with multiplication defined by $a\sim b=a+b+ab$ is isomorphic to the multiplicative group of nonzero real numbers, ...
0
votes
1answer
31 views

Check to see if my isomorphism is correct

Is multiplication modulo $10$ isomorphic to addition modulo $4$? $U(10) = \{1,3,7,9\}$, the identity is $1$, it is a cyclic group of order $4$, with generator $3$. $\Bbb Z_4 = \{0,1,2,3\}$, the ...
0
votes
2answers
702 views

How to find non-isomorphic trees?

"Draw all non-isomorphic trees with 5 vertices." I have searched the web and found many examples of the non-isomorphic trees with 5 vertices, but I can't figure out how they have come to their ...
0
votes
1answer
70 views

Isomorphism between two colored graphs

If there are two graphs whose shape is isomorphic to each other but whose combination of color used in each vertex is not isomorphic to that of other graph, how can I call their relationship? Should I ...
1
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1answer
96 views

Complete graph invariant

Does anybody know whether the multiset of the determinants (possibly together with the order of the submatrix they refer to) of all the principal minors of the (symmetric) adjacency matrix of a graph ...
0
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1answer
174 views

Checking for graph isomorphism by hand

I'm working through "A First Look at Graph Theory" by Clark & Holton, and in the first exercise, there are problems asking to check whether different graphs are isomorphic to each other. I find ...
0
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1answer
76 views

Determine all possible automorphisms of a graph

Let $G$ be the undirected graph whose vertex set is $\{a,b,c,d,e\}$ and edge set $\{ab, ae, bc, be, cd, ce, de\}$. The graph is drawn below: Let $V$ denote the set of vertices of the graph G ...
2
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2answers
3k views

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 ...
1
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1answer
48 views

Systematizing graph morphisms

Trying to systematize possible notions of graph morphisms I came about the following classification: A morphism $f$ which sends a graph $G$ to another graph $G'$ is – first of all – ...
2
votes
1answer
125 views

Proving that a graph is self complementary

I've been given the following adjacency matrix: $$\left(\begin{array}{cccccccc} 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 & 1 ...
1
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1answer
82 views

Can someone help me solve this misunderstanding please.

Is it a good way to determine isomorphic of two graphs by comparing their adjacent vertices? If their is a different then they are not isomorphic? If they have the same then they are isomorphic? Since ...
18
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4answers
2k views

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 deg(2) and 4 of deg(3) However, graph two has 2 ...
7
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2answers
160 views

Build graph with exactly n automorphisms

Construct graph with exactly distinct n automorphisms. For n $\geq$ 2. I wonder if we can just take an asymmetric graph, such as this one as building block.
2
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1answer
248 views

Finite Vertex-Transitive Planar game of Civilization?

If you have played games in the Civilization series, you will have noticed that the Earth is represented in a simplified and profoundly unsatisfying way. It is wrapped around the curve of a cylinder ...
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0answers
35 views

Rating and Scoring Graphs based on physical properties

i am busy working on a project related to L-systems. The basic idea is to generate graphs from these L-strings and rate them based on some physical traits, such as self similarity.... Is there any ...
3
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2answers
139 views

Number of non-isomorphic ways the following graph can be labelled

In how many non-isomorphic ways can the following graph be labelled? Ignore the numbers on the graph vertices. I got two different answers and I'm not sure which one of my reasoning is right: 1) ...
1
vote
1answer
338 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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2answers
202 views

Can someone intuitively explain how we can tell if two graphs are isomorphic?

I'm having a hard time understanding the explicit definition and was hoping someone could help me make a connection between the theory of isomorphism and the way it's actually applied (ex. how can we ...
0
votes
1answer
91 views

Graph isomorphic to symmetric group

Show why the symmetry group of the graph below is isomorphic to $S_3 \times S_2$. $S_3$ and $S_2$ are symmetric groups and $\times$ denotes direct product. *----------* /|\ | | | | ...
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votes
2answers
152 views

Finding an isomorphism between $(U\otimes V)^{\mathsf T}$ and $B(U \times V,\mathbb{R})$

Prove the isomorphism between $(U\otimes V)^{\mathsf T}$ and $B(U \times V,\mathbb{R})$, where $B$ is the collection of all bi-linear mappings. In order to do so, give a natural isomorphism between ...
1
vote
1answer
196 views

Generating non-isomorphic graph by adding two edges to a fixed graph

I am given a graph $G$ a fixed vertex $v \in V(G)$ and sets $S_1,S_2 \subseteq V(G).$ The problem I am currently studying requires to answer the following question Compute all non-isomorphic ...