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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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 ...
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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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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 ...
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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 ...
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Splitting a graph into two isomorphic parts

Say a graph $G$ has $2n$ vertices. I'd like to know if I can partition the vertices of $G$ into two parts $X$ and $Y$ such that $G[X]$ is isomorphic to $G[Y]$ ($G[S]$ denotes the subgraph of $G$ ...
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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 ...
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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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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 ...
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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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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 ...