Related papers: Uniqueness of exterior differentiation on locally …

We characterise the form of all simple, finite graphs for which the girth of the graph is equal to the circumference of the graph. We apply this to prove a bound on the number of edges in such a graph.

Consider a graph on the non-singular matrices over a finite field, in which two distinct non-singular matrices are joined by an edge whenever their sum is singular. We prove an upper bound for the independence number of this graph. As a…

We prove that the trace of nonlocal minimal graphs at points of stickiness is of class~$C^{1,\gamma}$. As a result, we show that boundary continuity implies boundary differentiability for nonlocal minimal graphs.

We establish new bounds on the minimum number of distinct eigenvalues among real symmetric matrices with nonzero off-diagonal pattern described by the edges of a graph and apply these to determine the minimum number of distinct eigenvalues…

We study the uniqueness of self-adjoint and Markovian extensions of the Laplacian on weighted graphs. We first show that, for locally finite graphs and a certain family of metrics, completeness of the graph implies uniqueness of these…

We prove that automorphic representations whose local components are certain small representations have multiplicity one. The proof is based on the multiplicity-one theorem for certain functionals of small representations, also proved in…

We show that, up to minor-equivalence, the Farey graph is the unique minor-minimal graph that is infinitely edge-connected but such that every two vertices can be finitely separated.

There are different definitions of ends in non-locally-finite graphs which are all equivalent in the locally finite case. We prove the compactness of the end-topology that is based on the principle of removing finite sets of vertices and…

We consider the uniqueness of solutions of ordinary differential equations where the coefficients may have singularities. We derive upper bounds on the the order of singularities of the coefficients and provide examples to illustrate the…

We give a sharp bound for the automorphism group of a cubic simple graph with a given number of vertices. For each number of vertices we give an explicit graph attaining the bound, and prove its uniqueness in special cases.

We show that if a group can be represented as a graph product of finite directly indecomposable groups, then this representation is unique.

We give characterizations for the failure of form uniqueness on weakly spherically symmetric graphs. The first characterization is in terms of the graph structure, the second involves the capacity of a Cauchy boundary. We also discuss the…

A recent development in graph-minor theory is to study local separators, vertex-sets that separate graphs locally but not necessarily globally. The local separators of a graph roughly correspond to the genuine separators of its local…

We introduce the notion of locally identifying coloring of a graph. A proper vertex-coloring c of a graph G is said to be locally identifying, if for any adjacent vertices u and v with distinct closed neighborhood, the sets of colors that…

In this paper, we consider the automorphisms of fine curve graphs restricted to continuously $k$-differentiable curves. We show that for closed surfaces with genus at least 2, they are induced by homeomorphisms of the surface.

We prove several criteria for quasi-isometry between non-locally-finite graphs and their structure trees. Results of M\"oller in \cite{moeller92ends2} for locally finite and transitive graphs are generalized. We also give a criterion which…

Mapping class groups of locally finite graphs are the analogue of those of infinite-type surfaces, and serve as a "big" version of $\text{Out}(F_n)$. In this paper, we investigate which of these mapping class groups have a dense conjugacy…

We show global uniqueness in the fractional Calder\'on problem with a single measurement and with data on arbitrary, possibly disjoint subsets of the exterior. The previous work \cite{GhoshSaloUhlmann} considered the case of infinitely many…

We study solutions of the generalized porous medium equation on infinite graphs. For nonnegative or nonpositive integrable data, we prove the existence and uniqueness of mild solutions on any graph. For changing sign integrable data, we…

We establish conditions under which the fundamental group of a graph of finite $p$-groups is necessarily residually $p$-finite. The technique of proof is independent of previously established results of this type, and the result is also…