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4
votes
1answer
259 views

How to find or constrain “particularly good” (two-sided) spectral expanders?

I'm new to graph theory, but a response to a question I asked a while ago introduced me to the concept of expander graphs. A k-regular graph (henceforth "graph") on n nodes has eigenvalues k = λ1 ≥ ...
0
votes
1answer
149 views

Efficient isomorphic subgraph matching with similarity scores

I'm a computer vision PhD student, and I'm looking for an efficient approximation to the following problem, which could end up helping in image to image matching. Failing that, pointers to relevant ...
8
votes
0answers
170 views

How many n/2-cycles can a cubic graph have

Given a simple cubic graph with $n$ vertices (which implies that $n$ is even), what is a good upper bound on the number of cycles of length $n/2$ it can have? A random cubic graph has ...
7
votes
0answers
141 views

Algorithms for computing the Resilience of Graphs

The definition of resilience with a graph $G$ w.r.t to a monotone property $\mathcal{P}$ is well known. (Global resilience) Let $\mathcal{P}$ be an increasing monotone property. The global ...
6
votes
0answers
345 views

Cliques of hyperedges

Suppose we have a graph, with multiple edges allowed. An edge-clique is a set $C$ of edges so that every two edges in $C$ share at least one endpoint. Note that any edge-clique falls into one of two ...
4
votes
0answers
412 views

Smallest matrix covered by many random n by n matrices

We say that a matrix $M$ can be covered by a (smaller) matrix $N$ if every entry in $M$ is contained in some submatrix of $M$ that exactly equals to $N$, up to reordering the rows and columns of $N$. ...
4
votes
0answers
311 views

An extremal problem for graphs having every edge contained in a 4-clique

This is a follow-up to Graphs with many triangles but few complete graphs on 4 vertices I'm looking for an upper bound for the difference between the number of edges and the number of 4-cliques in a ...
2
votes
0answers
136 views

The Turán problem for graphs with given chromatic number

The ordinary Turán problem for graphs asks, "Given a graph $H$, if $G$ is an $H$-free graph on $n$ vertices, what is the largest number of edges that $G$ can have?" As is well known, if $\chi(H) = r ...
2
votes
0answers
111 views

Bipartite independence number

Consider a balanced bipartite graph $G=(U,V,E)$, i.e., a bipartite graph with $|U|=|V|$. An independent set $I$ of $G$ is balanced if $|I \cap U| = |I \cap V|$. The bipartite independence number of ...
2
votes
0answers
113 views

Smallest size for an incomplete tournament with property $S_k$

By a well-known probabilistic argument due to Erdos, if $k>1$ is an integer then for all large enough $n$, there an asymmetric relation $R$ on $X=\lbrace 1,2, \ldots ,n \rbrace$ (i.e. $R \subseteq ...
1
vote
0answers
232 views

graphs with maximal number of paths of given length

Hi, For a given number of edges, the non directed graph which maximises the number of paths of length 2 is the quasi-star or the quasi-complete graph. Does anyone know : 1- what is the non directed ...