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We connect several notions relating the structural and dynamical properties of a graph. Among them are the topological entropy coming from the vertex shift, which is related to the spectral radius of the graph's adjacency matrix, the…
We define a growing model of random graphs. Given a sequence of nonnegative integers $\{d_n\}_{n=0}^\infty$ with the property that $d_i\leq i$, we construct a random graph on countably infinitely many vertices $v_0,v_1\ldots$ by the…
For a graph class $\mathcal{F}$, let $ex_{\mathcal{F}}(n)$ denote the maximum number of edges in a graph in $\mathcal{F}$ on $n$ vertices. We show that for every proper minor-closed graph class $\mathcal{F}$ the function…
Graph burning is a natural discrete graph algorithm inspired by the spread of social contagion. Despite its simplicity, some open problems remain steadfastly unsolved, notably the burning number conjecture, which says that every connected…
We show that large subsets of vector spaces over finite fields determine certain point configurations with prescribed distance structure. More specifically, we consider the complete graph with vertices as the points of $A \subseteq…
This work will appear as a chapter in a forthcoming volume titled "Topics in Probabilistic Graph Theory". A theory of scaling limits for random graphs has been developed in recent years. This theory gives access to the large-scale geometric…
We consider the problem of determining the maximum induced density of a graph H in any graph on n vertices. The limit of this density as n tends to infinity is called the inducibility of H. The exact value of this quantity is known only for…
The variation of the Randi\'c index $ R'(G) $ of a graph $G$ is defined by\ $R(G) = \sum_{uv \in E(G)}\frac 1{\max \{d(u) d(v)\}}$, where $d(u)$ is the degree of vertex $u$ and the summation extends over all edges $uv$ of $G$. Let $G(k,n)$…
A drawing of a graph is greedy if for each ordered pair of vertices u and v, there is a path from u to v such that the Euclidean distance to v decreases monotonically at every vertex of the path. The existence of greedy drawings has been…
We give sufficient conditions under which a random graph with a specified degree sequence is symmetric or asymmetric. In the case of bounded degree sequences, our characterisation captures the phase transition of the symmetry of the random…
Inspired by a concept in comparative genomics, we investigate properties of randomly chosen members of G_1(m,n,t), the set of bipartite graphs with $m$ left vertices, n right vertices, t edges, and each vertex of degree at least one. We…
We say that a vertex $v$ in a connected graph $G$ is decisive if the numbers of walks from $v$ of each length determine the graph $G$ rooted at $v$ up to isomorphism among all connected rooted graphs with the same number of vertices. On the…
Bounds on the minimum degree and on the number of vertices at- taining it have been much studied for finite edge-/vertex-minimally k- connected/k-edge-connected graphs. We give an overview of the results known for finite graphs, and show…
Let $G$ be a graph with a vertex set $V$. The graph $G$ is path-proximinal if there are a semimetric $d \colon V \times V \to [0, \infty[$ and disjoint proximinal subsets of the semimetric space $(V, d)$ such that $V = A \cup B$, and…
In this paper we obtain bounds on a very general class of distance-based topological indices of graphs, which includes the Wiener index, defined as the sum of the distances between all pairs of vertices of the graph, and most…
In a directed graph, the imbalance of a vertex is its outdegree minus its indegree. We characterize the sequences that are realizable as the sequence of imbalances of a simple directed graph. Moreover, a realization of a realizable sequence…
We give bounds on the L(2,1)-labeling number of a simple graph in terms of its order and its maximum degree. We also describe an infinite class of graphs of which the elements have the highest L(2,1)-labeling numbers in terms of their…
We apply Ramsey theoretic tools to show that there is a family of graphs which have tree-chromatic number at most~$2$ while the path-chromatic number is unbounded. This resolves a problem posed by Seymour.
It is well-known that in finite graphs, large complete minors/topological minors can be forced by assuming a large average degree. Our aim is to extend this fact to infinite graphs. For this, we generalise the notion of the relative end…
A path system $\mathcal{P}$ in a graph $G=(V,E)$ is said to be irreducible if there does not exist a partition $V= A\sqcup B$ such that $\mathcal{P}$ restricts to a path system on both $G[A]$ and $G[B]$. In this paper, we construct an…