# Tagged Questions

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### about the structure of components of tensor product if more than one bipartite graph is taken

I was reading about tensor product of graphs. We know that if we take tensor product of n graphs and want this product to be a connected graph then at most one graph should be bipartite. In the book ...
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### How can I prove that a particular family of graphs is integral?

I'm working with an infinite family of graphs that seems to always have all integral eigenvalues, and I'd like to find some way to prove that (if it's true). Call the graphs $G_{n,k}$ and define them ...
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### almost equitable partitions and spectra

If a graph $G$ has an equitable partition, then its charachteristic polynomial (for the adjacency matrix) has a divisor that can be seen as the characteristic polynomial (for the adjacency matrix) of ...
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### Upper bound on the number of vertex transitive graphs

Is there a known upper bound on the number of vertex transitive graphs on $n$ vertices?
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### On the existence of Graph Monomorphism

A graph monomorphism is an injective graph homomorphism. Determining existence of Graph monomorphism between graph pairs is computationally hard. Assume we talk only about classes of undirected ...
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### Graphs with many positive eigenvalues of their distance matrix

Let $G$ be a simple connected graph $D(G)$ its distance matrix and $n_{+}(G), n_{-}(G)$ the number of positive and negative eigenvalues of $D(G)$ respectively. We call a graph $G$ optimistic if ...
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### A traveling time problem

Given any undirected, connected and simple graph $G(V,E)$,each node of which is considered as a city. We call $j$ a neighbor of $i$ if $(i,j)\in E$. $N_i$ is the set of neighbors of $i$. $|V|=N$ ...
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### 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 ...
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### A question on graphs

Do there exist a family of graphs with the property: $$\left|\alpha\left({G \boxtimes \bar{G}}\right) - \alpha\left({\overline{G \boxtimes \bar{G}}}\right)\right| = O(\log(N_{G}))$$ where $G$ is the ...
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### vertex transitive graphs with 4p vertices with an imprimitivity block of length p and lexicographic product

Let $\Gamma$ be a vertex transitive graph with $4p$ vertices, where $p$ is an odd prime. Let $\Delta$ be an imprimitive block of length $p$ for the automorphism group of the graph. How can I prove ...
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### Graduate Schools for Graph Theory

I am a rising senior in a small liberal arts college, and I was wondering if anyone could suggest me good graduate schools for graph theory. My only exposure to graph theory has been the intro graph ...
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### Do right-profiles determine graphs up to isomorphism?

For graphs $G$ and $H$, let $h(G,H)$ denote the number of graph homomorphisms from $G$ to $H$. Fix some enumeration $G_1,G_2,\ldots$ of (isomorphism classes of) the set $\mathbf{D}$ of finite graphs, ...
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### Tutte polynomials, graph complements and degree sequences

Harary and Akiyama asked whether there exists a non self-complementary (SC) graph $G$ having the same chromatic polynomial as its complement. It was later shown that there indeed exist such graphs ...
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### Distance between vertices in a vertex transitive graphs. [closed]

Can anybody help me in finding out the distances between vertices in a vertex transitive graphs. Is there any specific formula to calculate distance between vertices in this graph. Thanks for your ...
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### What are some interesting almost equitable partitions which are not equitable?

There have been questions lately about almost equitable partitions in graphs, for example this one which provides the definition.) Every equitable partition is almost equitable. The converse is true ...
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### (The) missing Moore graph(s) - uniqueness

In the related literature one often sees the phrase "The missing Moore graph" which (to me) tacitly implies that the missing Moore graph (if exists) is unique. Is there a result of this type or is or ...
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### Bicycles and spanning trees of graphs

A spanning tree in a graph is a connected spanning subgraph with no cycles; it is well known that the number of spanning trees can be found by taking the determinant of a certain matrix related to the ...
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### Neat results from algebraic graph theory

Studying algebraic graph theory, I've stumbled across a wide range of results that I found pretty stunning and also useful. I'd like to share them here and ask for your favorite result from this area? ...
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### Graphs which are “distance-regular” with respect to a vertex (but not distance-regular)

A distance-regular graph (DRG) is, in essence, a graph $\Gamma$ of diameter $d$ for which there are integers $c_i, a_i, b_i, (0 \le i \le d)$ such that for all vertices $x$ of $\Gamma$ and for all ...
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### Cayley graphs and its subgraphs

I have two questions about Cayley graphs. Any answers will be appreciate. 1) Do we have any Cayley graph that has Petersen graph as its induced subgraph? 2) Suppose $Cay(G,S)$ be a Cayley graph that ...
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### Eigenvectors and partitions of graphs

Let G be an undirected graph with the node set $V$ and the Laplacian matrix $L$. Let $N(v)$ denote the neighbors of a node $v$ and $|N(v)|$ its degree. Then a partition $\pi=(V_1, V_2, \ldots, V_k)$ ...
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### Has anyone seen this graph?

I recently constructed the graph shown below in the process of investigating some problems regarding line graphs and homomorphisms, and then happened to see it on wikipedia. I was wondering if anyone ...
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### Is this statement about the real edge space of a graph known or trivial?

The statement is: ($u$ is a fixed node in a fixed graph $G$) $G$ is 3-connected if and only if the set of u-cycles span $\mathbb{R}^{E(G)}$. A u-cycle is a simple (no vertex repetitions) cycle in G ...
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### Graphs of order n with a Laplacian eigenvalue of multiplicity n-1.

I suspect this could be an easy one but I am not an expert in algebraic graph theory. Let $Q(G)$ define the Laplacian matrix for a simple graph $G$. It is well known that n is an eigenvalue of ...
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### Group of local complementation as a coxeter group

Can the group generated by local complementations, ${lc_i|i=1,\cdots,n}$ on simple graphs on $n$ vertices, be categorized as a coxeter group? After all these obey: \langle lc_i| ...
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### Local complementation group of simple graphs

This is my first time posting a question, so please excuse me for any incomplete or confusing descriptions. Let's assume we start with one simple graph(no multi-edges and no loops of a vertex to ...
I am looking at the automorphism group $G$ of a graph, represented as permutation matrices. The point in a proof I am trying to understand goes something like this: "For any permutation matrix $P$ ...