Related papers: On middle cube graphs
A recent result of one of the authors says that every connected subcubic bipartite graph that is not isomorphic to the Heawood graph has at least one, and in fact a positive proportion of its eigenvalues in the interval [-1,1]. We construct…
We study Hamiltonicity in random subgraphs of the hypercube $\mathcal{Q}^n$. Our first main theorem is an optimal hitting time result. Consider the random process which includes the edges of $\mathcal{Q}^n$ according to a uniformly chosen…
A certain signed adjacency matrix of the hypercube, which Hao Huang used last year to resolve the sensitivity conjecture, is closely related to the unique, 4-cycle free, 2-fold cover of the hypercube. We develop a framework in which this…
{\it A unit cube in $k$-dimension (or a $k$-cube) is defined as the cartesian product $R_1 \times R_2 \times ... \times R_k$, where each $R_i$ is a closed interval on the real line of the form $[a_i, a_i+1]$. The {\it cubicity} of $G$,…
We show how to test whether a graph with n vertices and m edges is a partial cube, and if so how to find a distance-preserving embedding of the graph into a hypercube, in the near-optimal time bound O(n^2), improving previous O(nm)-time…
In this paper we first obtain the spectrum of the folded hypercube in a new approach. Then we introduce a new family of graphs called the extended Hamming graph, denoted by $EH(n,2^n)$, which is constructed from the well-known Hamming graph…
Let m and r be two integers. Let G be a connected r-regular graph of order n and k an integer depending on m and r. For even kn, we find a best upper bound (in terms of r and m) on the third largest eigenvalue that is sufficient to…
In this paper, we propose a new type of graph, denoted as "embedded-graph", and its theory, which employs a distributed representation to describe the relations on the graph edges. Embedded-graphs can express linguistic and complicated…
The well-known middle levels problem is to find a Hammiltonian cycle in the graph induced from the binary Hamming graph $\cH_2(2k+1)$ by the words of weight $k$ or $k+1$. In this paper we define the $q$-analog of the middle levels problem.…
We give a complete characterisation of the cubic graphs with no eigenvalues in the interval $(-2,0)$. There is one thin infinite family consisting of a single graph on $6n$ vertices for each $n \geqslant 2$, and five ``sporadic'' graphs,…
We study the recognition complexity of subgraphs of k-connected planar cubic graphs for k = 1, 2, 3. We present polynomial-time algorithms to recognize subgraphs of 1- and 2-connected planar cubic graphs, both in the variable and fixed…
In this paper, we determine the set of all distinct eigenvalues of the line graph which is induced by the first and second layers of the hypercube $ Q_n $, $n>3$. We show that this graph has precisely five distinct eigenvalues and all of…
We introduce a new model for investigating spectral properties of quantum graphs, a quantum circulant graph. Circulant graphs are the Cayley graphs of cyclic groups. Quantum circulant graphs with standard vertex conditions maintain…
Define the middle layer graph as the graph whose vertex set consists of all bitstrings of length $2n+1$ that have exactly $n$ or $n+1$ entries equal to 1, with an edge between any two vertices for which the corresponding bitstrings differ…
The distance matrix of a graph $G$ is the matrix containing the pairwise distances between vertices. The distance eigenvalues of $G$ are the eigenvalues of its distance matrix and they form the distance spectrum of $G$. We determine the…
Let k be a natural number. We introduce k-threshold graphs. We show that there exists an O(n^3) algorithm for the recognition of k-threshold graphs for each natural number k. k-Threshold graphs are characterized by a finite collection of…
Given a graph G of order n and size m, let s(G)= sum|d(u)-2m/n|, where the sum is taken over all vertices u of G. We investigate upper and lower bounds on eigenvalues of G in terms of s(G).
The {\em Fibonacci cube} of dimension $n$, denoted as $\Gamma\_n$, is the subgraph of $n$-cube $Q\_n$ induced by vertices with no consecutive 1's. We study the maximum number of disjoint subgraphs in $\Gamma\_n$ isomorphic to $Q\_k$, and…
A $k$-orbit map is a map with its automorphism group partitioning the set of flags into $k$ orbits. Recently $k$-orbit maps were studied by Orbani\' c, Pellicer and Weiss, for $k \leq 4$. In this paper we use symmetry type graphs to extend…
We prove that a cubic graph with $m$ edges has an induced matching with at least $m/9$ edges. Our result generalizes a result for planar graphs due to Kang, Mnich, and M\"{u}ller (Induced matchings in subcubic planar graphs, SIAM J.…