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 formula; there are $n^{n-2}$ trees on n labelled vertices, so I’m guessing that this may come in handy. One idea I had […]

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 trees with 6 vertices, 7 vertices and so on).

From Lecture 2, Algebra and Computation by V. Arvind, (page2,3), I understood below passage- For our graph $G$, let $Aut(G) = H ≤ S_n$. We shall use Weilandt’s notation where $i^\pi$ denotes the image of i under$\pi$. In this notation, composition becomes simpler: $(i^\pi)^{\tau} = i^{\pi \tau}$ Define $H_i = \{\pi ∈ H : 1^\pi […]

A graph is said to be $k$-isoregular if for every subset $S$ of at most $k$ vertices the number common neighbors of the elements of $S$ depends only on the isomorphism type of the subgraph induced by $S$. Is there exists a $k$-iso-regular ($k \ge 3$) non-planar graph of degree at most three? If yes […]

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 $B$ […]

“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 answer. How exactly do you find how many non-isomorphic trees there are and what they look like? Thanks for your time

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 that isomorphic graphs can have same diagrams when represented on paper, but equal graphs also have same diagram on paper, if that is […]

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 $x \in G$. $A_x$ is the symmetric matrix of the graph $(G-x)$, where $C$ is the symmetric matrix of the graph created by vertices of $(G-x)$ which […]

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, graph two has $2$ simple circuits of length $3$ whereas graph one has only $1$ of length $3$. Is […]

I am using the book Graph Theory: A Problem Oriented Approach. One of the problems asks to list all the non-isomorphic graphs that can be made with four vertices. Fair enough, there are eleven total such non-isomorphic graphs. Then I grouped each graph with its corresponding complement, and there is one graph with three edges […]

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