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0
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4answers
110 views

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 ...
6
votes
2answers
177 views

adjacency matrix of a graph and lines on quartic surfaces

Suppose you are given a smooth quartic surface $X$ in $\mathbb P^3$. I would like to find an upper bound for the number of lines on $X$ in the case that there is no plane intersecting the curves in ...
1
vote
0answers
77 views

Triple Transitive Graphs

In the paper A Note on Triple Transitive Graphs, Cameron gives an almost-complete determination of triple transitive graphs with girth greater than $3$. In the remark at the end he says: "If, as ...
12
votes
3answers
282 views

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 ...
1
vote
1answer
95 views

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 ...
2
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1answer
198 views

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?
2
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0answers
203 views

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 ...
5
votes
1answer
284 views

On Turan's theorem

Turan's theorem provides minimum number of edges of a graph on $n$ vertices to surely contain a clique of a prescribed size. This has been generalized to regular graphs. What additional ...
5
votes
0answers
145 views

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 ...
1
vote
0answers
100 views

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 $i$. $|V|=N$ ...
4
votes
2answers
126 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 ...
1
vote
1answer
198 views

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 ...
0
votes
0answers
51 views

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 ...
1
vote
1answer
217 views

tutte's kirchhoff matrix and the matrix-tree theorem for digraphs

In his books, Tutte tells often of how he and his friends in Cambridge introduced a theory of non-symmetric electricity. His tales are always amusing and enjoyable, but often lack precise references; ...
2
votes
4answers
2k views

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 ...
4
votes
1answer
135 views

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, ...
12
votes
1answer
457 views

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 ...
1
vote
1answer
250 views

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 ...
1
vote
2answers
178 views

signs of eigenvalues

Let $\Gamma$ be a multiple edge free (di)graph (with or without loop). Let $A$ be its adjacency matrix. It is clear that if $\lambda^2$ is an eigenvalue of $A^2$, then $\lambda$ or $-\lambda$ is an ...
4
votes
3answers
348 views

automorphisms of graphs and finite permutation groups

I am interested in automorphisms of graphs and in using tools from permutation groups (especially such as in Wielandt's text on finite permutation groups, which I have been studying). What are some ...
2
votes
2answers
180 views

Is there any relation between automorphism group of a Cayley graph over a group and over its subgroup?

Let $\Gamma=Cay(G,S)$ be a Cayley graph over a group $G$, $H$ be a proper subgroup of $G$ and $\Sigma=Cay(H,T)$ where $S$ and $T$ are inversed-closed subsets of $G$ and $H$ not containing idendity, ...
2
votes
1answer
188 views

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 ...
1
vote
1answer
138 views

simple graphs of degree 16 with a semiregular normal subgroup isomorphic to the quaternion group $Q_8$

Is there any simple graph $\Gamma$ with 16 vertices with full automorphism group $G$ such that $H\cong Q_8$ be a semiregular normal subgroup of $G$?
2
votes
1answer
198 views

Strongly regular cayley graphs

Let $G$ and $Cay(A,S)$ be strongly regular graphs with the same parameters. Is it true that $G$ is a cayley graph?
0
votes
1answer
229 views

vertex-transitive graphs of order 10 with full automorphism group $A_5$ or $S_5$.

By a well-known result we know that a simply primitive permutation group of degree $2p$ where $p$ is a prime is $A_5$ or $S_5$ acting on 2-subsets of $\{1,\ldots,5\}$. The group has rank 3 and the ...
0
votes
1answer
62 views

imprimitive 2-blocks in connected Cayley (di)graphs of order twice a prime

Let $\Gamma=Cay(G,S)$ be a connected Cayley (di)graph over a group of order twice a prime and $\Sigma$ be a complete system of 2-blocks for $Aut(\Gamma)$. Let $K$ be the kernel of the action of ...
0
votes
2answers
198 views

Semiregular subgroups of automorphism group of cayley graphs

Let $\Gamma$ be a Cayley graph over group $K$ and $H$ be a semiregular subgroup of $Aut(\Gamma)$ with two orbits. Then $|K|=2|H|$. Is there any other relation between $H$ and $K$ in general? What ...
4
votes
1answer
452 views

Algebraic characterisation of directed acyclic graphs

Any characterization based on the adjacency matrix for directed acyclic graphs (DAG)? An undirected graph could be simply characterized by saying that its adjacency matrix is symmetric. What about a ...
6
votes
1answer
317 views

(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 ...
7
votes
0answers
325 views

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 ...
3
votes
2answers
525 views

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? ...
4
votes
1answer
259 views

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 ...
7
votes
4answers
1k views

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 ...
6
votes
3answers
1k views

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 $\pi=(V_1, V_2, \ldots, V_k)$ ...
5
votes
1answer
877 views

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 ...
3
votes
3answers
476 views

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 ...
3
votes
2answers
387 views

Lovasz function equality - combinatoric graph theory.

Hello everyone, I was wondering how to prove the following equality: $\theta(G+H)=\theta(G)+\theta(H)$ where $G$ and $H$ are graphs and $\theta$ is the Lovasz Theta function. correction: I ...
2
votes
1answer
203 views

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 ...
2
votes
0answers
220 views

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: \begin{equation} \langle lc_i| ...
6
votes
1answer
448 views

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 ...
5
votes
0answers
396 views

“sum over labelings” representations of graph polynomials

It seems that there's a general way to go from "recursive" definition of a graph polynomials to "subset expansion" formulas. Furthermore, polynomials with subset expansion formulas often have a ...
0
votes
3answers
306 views

Morphisms between representations

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 $P$ ...