Related papers: Aldous property for full-flag Johnson graphs
In recent work on equiangular lines, Jiang, Tidor, Yuan, Zhang, and Zhao showed that a connected bounded degree graph has sublinear second eigenvalue multiplicity. More generally they show that there cannot be too many eigenvalues near the…
This paper addresses the challenge of spectral analysis and structural investigation for graphs that are not distance-regular, where computing the spectrum using standard methods based on equitable and orbit partitions can be complex. Our…
We call a set $\mathcal S$ of graphs an "even subdivison-factor" of a cubic graph $G$ if $G$ contains a spanning subgraph $H$ such that every component of $H$ has an even number of vertices and is a subdivision of an element of $\mathcal…
Let the group $G$ act transitively on the finite set $\Omega$, and let $S \subseteq G$ be closed under taking inverses. The Schreier graph $Sch(G \circlearrowleft \Omega,S)$ is the graph with vertex set $\Omega$ and edge set $\{…
The paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics, nanotechnology, optics, and…
We study Schr\"odinger operators on metric and discrete decorated graphs. The values taken by the integrated density of states (IDS) on spectral gaps are called gap labels. A natural question is which gap labels can occur. We answer this…
We locate gaps in the spectrum of a Hamiltonian on a periodic cuboidal (and generally hyperrectangular) lattice graph with $\delta$ couplings in the vertices. We formulate sufficient conditions under which the number of gaps is finite. As…
Let $a,b$ be two positive integers such that $a \le b$ and $a \equiv b$ (mod $2$). We say that a graph $G$ has an $(a,b)$-parity factor if $G$ has a spanning subgraph $F$ such that $d_{F}(v) \equiv b$ (mod $2$) and $a \le d_{F}(v) \le b$…
We enumerate factorisations of the complete bipartite graph into spanning semiregular graphs in several cases, including when the degrees of all the factors except one or two are small. The resulting asymptotic behaviour is seen to…
Hadwiger's Conjecture states that every graph with chromatic number $k$ contains a complete graph on $k$ vertices as a minor. This conjecture is a tremendous strengthening of the Four-Colour Theorem and is regarded as one of the most…
Regular and distance-regular characterizations of general graphs are well-known. In particular, the spectral excess theorem states that a connected graph G is distance-regular if and only if its spectral excess (a number that can be…
Sidorenko's conjecture asserts that every bipartite graph $H$ has the property that, for any host graph $G$, the homomorphism density from $H$ to $G$ is asymptotically at least as large as in a quasirandom graph with the same edge density…
Khovanov introduced a bigraded cohomology theory of links whose graded Euler characteristic is the Jones polynomial. The theory was subsequently applied to the chromatic polynomial of graph, resulting in a categorification known as the…
In this paper, we show that every connected signed graph with smallest eigenvalue strictly greater than $-2$ and large enough minimum degree is switching equivalent to a complete graph. This is a signed analogue of a theorem of Hoffman. The…
The slope conjecture proposed by Garoufalidis asserts that the Jones slopes given by the sequence of degrees of the colored Jones polynomials are boundary slopes. We verify the slope conjecture for graph knots, i.e. knots whose Gromov…
Generally speaking, `almost distance-regular' graphs share some, but not necessarily all, of the regularity properties that characterize distance-regular graphs. In this paper we propose two new dual concepts of almost distance-regularity,…
In 1994, it was conjectured by Fan and Raspaud that every simple bridgeless cubic graph has three perfect matchings whose intersection is empty. In this paper we answer a question recently proposed by Mkrtchyan and Vardanyan, by giving an…
Let $a\leq b$ be two positive integers. We say that a graph $G$ has all $[a,b]$-factors if it has an $h$-factor for every function $h: V(G)\rightarrow \mathbb{Z}^+$ such that $a\le h(v) \le b$ for all $v\in V(G)$ and $\sum_{v\in…
Recently, Huang gave a very elegant proof of the Sensitivity Conjecture by proving that hypercube graphs have the following property: every induced subgraph on a set of more than half its vertices has maximum degree at least $\sqrt{d}$,…
We offer a new method for proving that the maximal eigenvalue of the normalized graph Laplacian of a graph with $n$ vertices is at least $\frac{n+1}{n-1}$ provided the graph is not complete and that equality is attained if and only if the…