**3**

votes

**2**answers

133 views

### Characterizing graphs whose Incidence Matrix has the all ones vector in its row span

Suppose we have a simple connected graph $G=(V,E)$. Then let $A$ be its $|E|\times |V|$ incidence matrix. Here I am considering the unoriented incidence matrix. I want to known when the row span of $A$...

**6**

votes

**2**answers

478 views

### A question about (unicity of certain cycles in a Cayley graph of a) symmetric group

Let $S=\{(1,2),(1,2,3,\ldots,n),(1,2,3,\ldots,n)^{-1}=(1,n\ldots,2)\}$ be a subset of the symmetric group $S_n$. We know that $(1,2,\ldots,n)(1,2)=(2,3,\ldots,n)$, and thus $$[(1,2,\ldots,n)(1,2)]^{n-...

**3**

votes

**3**answers

151 views

### Graphs cospectral with Cayley graphs

Let $G$ be a Cayley graph, and $H$ a graph cospectral with $G$. Must $H$ be a Cayley graph? Does a counterexample exist? If $G$ is a circulant graph, does a counterexample exist?

**0**

votes

**0**answers

50 views

### normal sets and conjugate generating sets of $S_n$

In this arXiv paper (p. 13), Steinhardt defines a normal set in $S_n$ as follows:
Definition: A split set of more than two cycles generating $S_n$ is said to be normal if any element is adjacent to ...

**6**

votes

**3**answers

244 views

### Numerical invariants for a graph or its complement that are bounded by some constant

I'm looking for numerical graph invariants that are bounded by a constant either for a graph $G$ or its complement $\bar{G}$. (The complement graph $\bar{G}$ has the same set of vertices as $G$ but ...

**0**

votes

**0**answers

112 views

### Ramanujan graphs and stable real polynomials

I have a problem with the lemma 6.5 and the theorems 5.1, 6.6 in the paper of Adam Marcus, Daniel Spielman and Nikhil Srivastava. http://arxiv.org/pdf/1304.4132.pdf
I do not understand how they use ...

**1**

vote

**0**answers

35 views

### Atomic parts of lexicographic products of vertex-transitive graphs

Suppose $H_1$ and $H_2$ are connected, vertex-transitive graphs, $H_1$ is not the complete graph, and $|V(H_2)| \ge 2$. Then, the lexigraphic product $G=H_1 \circ H_2$ is vertex-transitive, $0 < \...

**3**

votes

**1**answer

121 views

### Vertex-connectivity of connected, vertex-transitive graphs without $K_4$ is maximum possible

A graph is said to have optimal vertex connectivity if its vertex connectivity equals its minimum degree. According to this arXiv preprint, it was shown by Mader in (Arch. Math., 1970) and (Math. Ann.,...

**0**

votes

**1**answer

42 views

### Number of $k$-walks containing a vertex in an unweighted multigraph

Let $G = (V,E,W)$ be a weighted graph, where each edge $e = (v_i,v_j)$ has weight $w_{ij} \in \mathbb Z^+ \cup \{0\}$. By replacing $e$ with $w_{ij}$ copies of unweighted multiedges, a weighted graph $...

**3**

votes

**0**answers

103 views

### The degree/diameter problem for even girth graphs starting with upper bound

I posted this on stackexchange but due to a lack of response there I am posting here.
Let $G$ be a graph with girth $g$, minimal degree $\delta$, maximal degree $\Delta$, and diameter $D$. Define $$...

**6**

votes

**3**answers

256 views

### Generating (or availability of) large strongly regular graphs

Are there collections of already generated large strongly regular graphs available to download? By large I mean $n \geq 200$ where $n$ is the number of vertices. I found Ted Spence's page on srgs, ...

**5**

votes

**0**answers

103 views

### Diameter of the modified bubble-sort graph

The modified bubble-sort graph is the Cayley graph $Cay(S_n,S)$ of $S_n$ generated by $n$ cyclically adjacent transpositions. Thus $S = \{ (1,2),(2,3),\ldots,(n,1)\}$. I was wondering whether the ...

**6**

votes

**1**answer

134 views

### Relationship of Weisfeiler-Lehman algorithm to weak isomorphism of coherent algebras

A coherent algebra is a matrix algebra (over $\mathbb{C}$) closed under conjugate transpose and Schur (entrywise) product, and that contains the identity matrix $I$ and all ones matrix $J$. Given ...

**0**

votes

**0**answers

154 views

### Cayley graphs with special subgraphs and some related problems

I asked some questions about finite Cayley graphs with special type of subgraphs which has been answered by Dear Prof. Godsil. It can be seen in the MO page with address:
Cayley graphs and its ...

**0**

votes

**1**answer

116 views

### Graph automorphisms that preserve independent sets [closed]

Let $G=(V,E)$ be a graph and $\mathrm{Ind}(G)$ be the collection of its independent sets.
We call a graph automorphism $f:V \to V$ of $G$ good if it is non-trivial and $f(\mathrm{Ind}(G))=\mathrm{...

**1**

vote

**1**answer

126 views

### Linear algebra formulation for colored node graph isomorphism

(Please see a few paragraphs below by what I mean by “colored node graph isomorphism”.)
Some basic definitions for completeness:
Given two graphs $G_1=(V_1, E_1)$ and $G_2=(V_2, E_2)$ the graph ...

**5**

votes

**1**answer

225 views

### When is the direct product of two graph cores itself a core?

A graph homomorphism $f$ is a function $f : V(X) \to V(Y)$ such that if $uv \in E(X)$, then $f(u)f(v) \in E(Y)$. If such an $f$ exists, write $X \to Y$. $X$ and $Y$ are hom-equivalent if $X \to Y$ and ...

**5**

votes

**2**answers

229 views

### Condition(s) for the full autormophism group $\operatorname{Aut}(C(G, S))$ of the Cayley graph of $G$ to be isomorphic to $G$

If $\Gamma = C(G, S)$ is the (undirected) Cayley graph of a finite group $G$ with generating set $S$, then $G \le \operatorname{Aut}(\Gamma)$, the "full" automorphism group of $\Gamma$.
When is ...

**4**

votes

**0**answers

177 views

### Sets of spreads in graphs

Let $G$ be a graph. A $k$-spread is a set of cliques of order $k$ which partition the vertex set (so $k|n$, where $n$ is the number of vertices).
A partial $k$-resolution of $G$ is a set of pairwise ...

**4**

votes

**1**answer

202 views

### Structure of the stabilizer of a vertex-neighborhood of a vertex-transitive graph

Given a simple, undirected graph and a vertex $v$ of the graph, let $L_v$ denote the set of automorphisms of the graph that fixes the vertex $v$ and each of its neighbors. When the graph is vertex-...

**6**

votes

**1**answer

298 views

### Do perfect matching(s) have signatures in the graph eigenvalues?

If the edges of a bipartite graph are such that they can be seen as a disjoint union of perfect matchings then will this somehow reflect in the eigenvalues of the Laplacian?
It would be helpful to ...

**4**

votes

**2**answers

242 views

### Graphs whose degree vectors coincide for all powers of their adjacency matrices

Let symmetric $A,B \in \{0, 1\}^{n \times n}$ denote the adjacency matrices of two simple graphs. Further let $\mathbf{1}$ denote the all-one-vector.
Now assume that $A^k \mathbf{1} = B^k \mathbf{1}$ ...

**9**

votes

**2**answers

266 views

### Vertex-primitive graphs with two vertices having almost the same neighbourhood

Hypothesis: Let $\Gamma$ be a vertex-primitive graph with two vertices $u$ and $v$ such that $$|N(u) \cap N(v)|=|N(v)|-1$$
Question: Is it true that $\Gamma$ must either be a complete graph or have ...

**0**

votes

**1**answer

75 views

### edge transitivity and edge deletion

Let G be a graph which has the following properties:
1) For every $e_1,e_2 \notin E(G)$, $G \cup e_1 \cong G \cup e_2$
2) For every $e_1,e_2 \in E(G)$, $G\setminus e_1 \cong G\setminus e_2$
i.e. ...

**0**

votes

**4**answers

251 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

**2**answers

275 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 ...

**2**

votes

**0**answers

101 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

**3**answers

321 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

**1**answer

226 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**

votes

**1**answer

242 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**

votes

**0**answers

304 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 ...

**6**

votes

**1**answer

397 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

**0**answers

237 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 $n_{+}...

**1**

vote

**0**answers

115 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

**2**answers

147 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**

votes

**1**answer

212 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

**0**answers

58 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 ...

**2**

votes

**1**answer

357 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; ...

**1**

vote

**5**answers

4k views

### Graduate Schools for Graph Theory [closed]

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

**1**answer

151 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

**1**answer

582 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

**1**answer

416 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

**2**answers

191 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 ...

**5**

votes

**3**answers

482 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

**2**answers

345 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

**1**answer

237 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 ...

**2**

votes

**1**answer

159 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

**1**answer

256 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

**1**answer

294 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

**1**answer

73 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 $Aut(\...