Related papers: The Triangle Friendship Paradox

We study the metric dimension (strong and weak) of infinite graphs. In particular, our main interest is characterizing infinite graphs with finite dimension. Our main results: (1) graphs with more than one end have infinite strong…

Although community detection has drawn tremendous amount of attention across the sciences in the past decades, no formal consensus has been reached on the very nature of what qualifies a community as such. In this article we take an…

A celebrated result of Mantel shows that every graph on $n$ vertices with $\lfloor n^2/4 \rfloor + 1$ edges must contain a triangle. A robust version of this result, due to Rademacher, says that there must in fact be at least $\lfloor n/2…

In this thesis we consider ordered graphs (that is, graphs with a fixed linear ordering on their vertices). We summarize and further investigations on the number of edges an ordered graph may have while avoiding a fixed forbidden ordered…

We propose a new distribution-free model of social networks. Our definitions are motivated by one of the most universal signatures of social networks, triadic closure---the property that pairs of vertices with common neighbors tend to be…

We count the asymptotic number of triangles in uniform random graphs where the degree distribution follows a power law with degree exponent $\tau\in(2,3)$. We also analyze the local clustering coefficient $c(k)$, the probability that two…

We consider the edge-triangle model, a two-parameter family of exponential random graphs in which dependence between edges is introduced through triangles. In the so-called replica symmetric regime, the limiting free energy exists together…

A triple of vertices in a graph is a \emph{frustrated triangle} if it induces an odd number of edges. We study the set $F_n\subset[0,\binom{n}{3}]$ of possible number of frustrated triangles $f(G)$ in a graph $G$ on $n$ vertices. We prove…

The celebrated Mantel's theorem states that any triangle-free graph on $n$ vertices contains at most $\left\lfloor n^2/4\right\rfloor$ edges. It is natural to ask how many triangles must exist in a graph with more than $\left\lfloor…

The number of triangles in a graph is useful to deduce a plethora of important features of the network that the graph is modeling. However, finding the exact value of this number is computationally expensive. Hence, a number of…

Recent work on the structure of social networks and the internet has focussed attention on graphs with distributions of vertex degree that are significantly different from the Poisson degree distributions that have been widely studied in…

A drawing of a graph in the plane is a thrackle if every pair of edges intersects exactly once, either at a common vertex or at a proper crossing. Conway's conjecture states that a thrackle has at most as many edges as vertices. In this…

We deal with a general preferential attachment graph model with multiple type edges. The types are chosen randomly, in a way that depends on the evolution of the graph. In the $N$-type case, we define the (generalized) degree of a given…

We investigate the joint distribution of the vertex degrees in three models of random bipartite graphs. Namely, we can choose each edge with a specified probability, choose a specified number of edges, or specify the vertex degrees in one…

In this note, we give answers to three questions from the paper [A. Das, Triameter of graphs, Discuss. Math. Graph Theory, 41 (2021), 601--616]. Namely, we obtain a tight lower bound for the triameter of trees in terms of order and number…

For a family $\mathcal{F}$ of graphs, let $ex(n,\mathcal{F})$ denote the maximum number of edges in an $n$-vertex graph which contains none of the members of $\mathcal{F}$ as a subgraph. A longstanding problem in extremal graph theory asks…

Multi-edge networks capture repeated interactions between individuals. In social networks, such edges often form closed triangles, or triads. Standard approaches to measure this triadic closure, however, fail for multi-edge networks,…

We give a sharp bound on the number of triangles in a graph with fixed number of edges. We also characterize graphs that achieve the maximum number of triangles. Using the upper bound on number of triangles, we prove that if $G$ is a…

We study triangle decompositions of graphs. We consider constructions of classes of graphs where every edge lies on a triangle and the addition of the minimum number of multiple edges between already adjacent vertices results in a strongly…

Triangles are abundant in real-world networks but rare in standard null models for sparse graphs. Existing explanations typically rely on explicit triadic closure mechanisms or geometry-based connection rules. We propose an alternative…