We study the large-volume asymptotics of the sum of power-weighted edge lengths $\sum_{e \in E}|e|^\alpha$ in Poisson-based spatial random networks. In the regime $\alpha > d$, we provide a set of sufficient conditions under which the upper-large-deviation asymptotics are characterized by a condensation phenomenon, meaning that the excess is caused by a negligible portion of Poisson points. Moreover, the rate function can be expressed through a concrete optimization problem. This framework encompasses in particular directed, bidirected, and undirected variants of the k-nearest-neighbor graph, as well as suitable $\beta$-skeletons.

Let $G=(V,E)$ be a countable graph. The Bunkbed graph of $G$ is the product graph $G \times K_2$ , which has vertex set $V\times \{0,1\}$ with “horizontal” edges inherited from $G$ and additional “vertical” edges connecting $(w,0)$ and $(w,1)$ for each $w \in V$ . Kasteleyn’s Bunkbed conjecture states that for each $u,v \in V$ and $p\in [0,1]$ , the vertex $(u,0)$ is at least as likely to be connected to $(v,0)$ as to $(v,1)$ under Bernoulli- $p$ bond percolation on the bunkbed graph. We prove that the conjecture holds in the $p \uparrow 1$ limit in the sense that for each finite graph $G$ there exists $\varepsilon (G)\gt 0$ such that the bunkbed conjecture holds for $p \geqslant 1-\varepsilon (G)$ .

Let R be a commutative ring with identity which is not an integral domain. An ideal I of R is called an annihilating ideal if there exists $r\in R- \{0\}$ such that $Ir=(0)$ . The total graph of nonzero annihilating ideals of R is the graph $\Omega (R)$ whose vertices are the nonzero annihilating ideals of R and two distinct vertices $I,J$ are joined if and only if $I+J$ is also an annihilating ideal of R. We study the strong metric dimension of $\Omega (R)$ and evaluate it in several cases.

An emerging technique in image segmentation, semi-supervised learning and general classification problems concerns the use of phase-separating flows defined on finite graphs. This technique was pioneered in Bertozzi and Flenner (2012, Multiscale Modeling and Simulation10(3), 1090–1118), which used the Allen–Cahn flow on a graph, and was then extended in Merkurjev et al. (2013, SIAM J. Imaging Sci.6(4), 1903–1930) using instead the Merriman–Bence–Osher (MBO) scheme on a graph. In previous work by the authors, Budd and Van Gennip (2020, SIAM J. Math. Anal.52(5), 4101–4139), we gave a theoretical justification for this use of the MBO scheme in place of Allen–Cahn flow, showing that the MBO scheme is a special case of a ‘semi-discrete’ numerical scheme for Allen–Cahn flow. In this paper, we extend this earlier work, showing that this link via the semi-discrete scheme is robust to passing to the mass-conserving case. Inspired by Rubinstein and Sternberg (1992, IMA J. Appl. Math.48, 249–264), we define a mass-conserving Allen–Cahn equation on a graph. Then, with the help of the tools of convex optimisation, we show that our earlier machinery can be applied to derive the mass-conserving MBO scheme on a graph as a special case of a semi-discrete scheme for mass-conserving Allen–Cahn. We give a theoretical analysis of this flow and scheme, proving various desired properties like existence and uniqueness of the flow and convergence of the scheme, and also show that the semi-discrete scheme yields a choice function for solutions to the mass-conserving MBO scheme.

A sequence of graphs is FO-convergent if the probability of satisfaction of every first-order formula converges. A graph modeling is a graph, whose domain is a standard probability space, with the property that every definable set is Borel. It was known that FO-convergent sequence of graphs do not always admit a modeling limit, but it was conjectured that FO-convergent sequences of sufficiently sparse graphs have a modeling limits. Precisely, two conjectures were proposed:

1. 1. If a FO-convergent sequence of graphs is residual, that is if for every integer d the maximum relative size of a ball of radius d in the graphs of the sequence tends to zero, then the sequence has a modeling limit.

2. 2. A monotone class of graphs ${\cal C}$ has the property that every FO-convergent sequence of graphs from ${\cal C}$ has a modeling limit if and only if ${\cal C}$ is nowhere dense, that is if and only if for each integer p there is $N\left( p \right)$ such that no graph in ${\cal C}$ contains the pth subdivision of a complete graph on $N\left( p \right)$ vertices as a subgraph.

In this article we prove both conjectures. This solves some of the main problems in the area and among others provides an analytic characterization of the nowhere dense–somewhere dense dichotomy.

Every graphon defines a random graph on any given number n of vertices. It was known that the graphon is random-free if and only if the entropy of this random graph is subquadratic. We prove that for random-free graphons, this entropy can grow as fast as any subquadratic function. However, if the graphon belongs to the closure of a random-free hereditary graph property, then the entropy is O(n log n). We also give a simple construction of a non-step-function random-free graphon for which this entropy is linear, refuting a conjecture of Janson.

Hu et al. [“A boundary problem for group testing”, SIAM J. Algebraic Discrete Meth.2 (1981), 81–87] conjectured that the minimax test number to find d defectives in 3d items is 3d−1, a surprisingly difficult combinatorial problem about which very little is known. In this article we state three more conjectures and prove that they are all equivalent to the conjecture of Hu et al. Notably, as a byproduct, we also obtain an interesting upper bound for M(d,n).

We classify linear weighted graphs up to the blowing-up and blowing-down operations which are relevant for the study of algebraic surfaces.

We define a subgraph of the zero divisor graph of a ring, associated to the ring idempotents. We study its properties and prove that for large classes of rings the connectedness of the graph is equivalent to the indecomposability of the ring and in those cases we also calculate the graph’s diameter.

This paper studies the Laplacian operator on a metrized graph, and its spectral theory.

We give a unified treatment of several fixed-point type theorems by using the concept of dismantlability, extended from ordered sets to arbitrary graphs. For a graph G and a vertex x of G we let NG(X) denote the set of neighbours of x in G. We say that x is a subdominant vertex of G if there is a vertex y of G, distinct from x, such that NG(x)∪{x} ⊆ NG(y)∪{y}. If G has n vertices we say that G is dismantlable if the vertices of G can be listed as x1, x2, ..., xi,..., xn such that, for all i = 1,2,..., n— 1, xi is a subdominant vertex of the graph Gi = G — {xj : j < i}

The intricacy of the problem of (2s + 1 )-factorizating Kn is determined. Some generalizations are also given.

D. R. Woodall has introduced closely-related notions of Menger and König duals which can be applied to a broad range of combinatorial contexts. The present paper considers these two notions for finite ground sets in terms of syntactic duality principles. Specific graph-theoretic interpretations are cited.

A star is a connected graph in which every node but possibly one has valency 1. Let G be a graph and C a spanning subgraph of G in which every component is a star. With each component α of C let us associate a weight wα. Let Пα wα be the weight associated with the entire subgraph G the star polynomial of G is ΣПα wα where the summation is taken over all spanning subgraphs of G consisting of stars. In this paper an algorithm for finding star polynomials of graphs is given. The star polynomials of various classes of graphs are then found, and some results about node-disjoint decomposition of complete graphs and complete bipartite graphs are deduced.