Related papers: Graph Theory
This paper contains the proofs of Theorems 2 and 3 of the article entitled Random Electrical Networks on Complete Graphs, written by the same authors and published in the Journal of the London Mathematical Society, vol. 30 (1984), pp.…
In this paper, we provide the notions of connection $1$-forms and curvature $2$-forms on graphs. We prove a Weitzenb\"ock formula for connection Laplacians in this setting. We also define a discrete Yang-Mills functional and study its…
For a graph $G$, denote by $t_r(G)$ (resp. $b_r(G)$) the maximum size of a $K_r$-free (resp. $(r-1)$-partite) subgraph of $G$. Of course $t_r(G) \geq b_r(G)$ for any $G$, and Tur\'an's Theorem says that equality holds for complete graphs.…
Theory for open curves over a local field. After introducing the reciprocity map, we determine the kernel and the cokernel of this map. In addition to this, the Pontrjagin dual of the reciprocity map is also investigated. This gives the one…
Graphs are structured data that models complex relations between real-world entities. Heterophilic graphs, where linked nodes are prone to be with different labels or dissimilar features, have recently attracted significant attention and…
There are three approaches in the current social network analysis study: Graph Representation, Content Mining, and Semantic Analysis. Graph Representation has been used for analyzing social network topology, structural modeling,…
For each commutative, graded algebra with finite dimension in each degree, we construct a graded cohomology theory for graphs whose graded Euler characteristic is the chromatic polynomial of the graph. This extends our previous work which…
Many graph properties (e.g., connectedness, containing a complete subgraph) are known to be difficult to check. In a decision-tree model, the cost of an algorithm is measured by the number of edges in the graph that it queries. R. Karp…
The adaptive processing of graph data is a long-standing research topic which has been lately consolidated as a theme of major interest in the deep learning community. The snap increase in the amount and breadth of related research has come…
We give yet another proof of the list-color version of Brooks' theorem that is due, independently, to Vizing and to Erd\H{o}s, Rubin and Taylor, via a famous theorem of Dirac on chordal graphs.
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…
In this paper we study the class of graphs $G_{m,n}$ that have the same degree sequence as two disjoint cliques $K_m$ and $K_n$, as well as the class $\overline G_{m,n}$ of the complements of such graphs. We establish various properties of…
We study a family of C*-algebras generalizing both Katsura algebras and certain algebras introduced by Nekrashevych in terms of self-similar groups.
These are lecture notes that are based on the lectures from a class I taught on the topic of Randomized Linear Algebra (RLA) at UC Berkeley during the Fall 2013 semester.
The application of graph theory to model the complex structure and function of the brain has shed new light on its organization and function, prompting the emergence of network neuroscience. Despite the tremendous progress that has been…
We show that a random graph studied by Ioffe and Levit is an example of an inhomogeneous random graph of the type studied by Bollobas, Janson and Riordan, which enables us to give a new, simple, proof of their result on a phase transition.
A speculative overview of a future topic of research. The paper is a collection of ideas concerning two related areas: 1) Graph computation machines ("computing with graphs"). This is the class of models of computation in which the state of…
Multiparticle systems on complicated metric graphs might have many applications in physics, biology and social life. But the corresponding science still does not exist. Here we start it with simplest examples where there is quadratic…
Machine learning on graphs, especially using graph neural networks (GNNs), has seen a surge in interest due to the wide availability of graph data across a broad spectrum of disciplines, from life to social and engineering sciences. Despite…
We study the random graph obtained by random deletion of vertices or edges from a random graph with given vertex degrees. A simple trick of exploding vertices instead of deleting them, enables us to derive results from known results for…