arXiv subjects

Babak Miraftab

Publications and source records attributed to Babak Miraftab.

30 records · Page 2Linked to original sources

arXiv2023

On Matrix Product Factorization of graphs

In this paper, we explore the concept of the ``matrix product of graphs," initially introduced by Prasad, Sudhakara, Sujatha, and M. Vinay. This operation involves the multiplication of adjacency matrices of two graphs with assigned labels, resulting in a weighted digraph. Our primary focus is on identifying graphs that can be expressed as the graphical matrix product of two other graphs. Notably, we establish that the only complete graph fitting this framework is $K_{4n+1}$, and moreover the factorization is not unique. In addition, the only complete bipartite graph that can be expressed as the graphical matrix product of two other graphs is $K_{2n,2m}$ Furthermore, we introduce several families of graphs that exhibit such factorization and, conversely, some families that do not admit any factorization such as wheel graphs, friendship graphs, hypercubes and paths.

math.CO↗

arXiv2023

On vertex-transitive graphs with a unique hamiltonian cycle

A graph is said to be uniquely hamiltonian if it has a unique hamiltonian cycle. For a natural extension of this concept to infinite graphs, we find all uniquely hamiltonian vertex-transitive graphs with finitely many ends, and also discuss some examples with infinitely many ends. In particular, we show each nonabelian free group $F_n$ has a Cayley graph of degree $2n + 2$ that has a unique hamiltonian circle. (A weaker statement had been conjectured by A. Georgakopoulos.) Furthermore, we prove that these Cayley graphs of $F_n$ are outerplanar.

math.CO↗

arXiv2022

Arc-disjoint hamiltonian paths in Cartesian products of directed cycles

We show that if $C_1$ and $C_2$ are directed cycles (of length at least two), then the Cartesian product $C_1 \Box C_2$ has two arc-disjoint hamiltonian paths. (This answers a question asked by J. A. Gallian in 1985.) The same conclusion also holds for the Cartesian product of any four or more directed cycles (of length at least two), but some cases remain open for the Cartesian product of three directed cycles. We also discuss the existence of arc-disjoint hamiltonian paths in $2$-generated Cayley digraphs on (finite or infinite) abelian groups.

math.CO↗

arXiv2021

Splitting groups with cubic Cayley graphs of connectivity two

A group $G$ splits over a subgroup $C$ if $G$ is either a free product with amalgamation $A \underset{C}{\ast} B$ or an HNN-extension $G=A \underset{C}{\ast} (t)$. We invoke Bass-Serre theory and classify all infinite groups which admit cubic Cayley graphs of connectivity two in terms of splittings over a subgroup.

math.CO↗

arXiv2020

Canonical trees of tree-decompositions

We prove that every graph has a canonical tree of tree-decompositions that distinguishes all principal tangles (these include the ends and various kinds of large finite dense structures) efficiently. Here `trees of tree-decompositions' are a slightly weaker notion than `tree-decompositions' but much more well-behaved than `tree-like metric spaces'. This theorem is best possible in the sense that we give an example that `trees of tree-decompositions' cannot be strengthened to `tree-decompositions' in the above theorem. This implies results of Dunwoody and Krön as well as of Carmesin, Diestel, Hundertmark and Stein. Beyond that for locally finite graphs our result gives for each $k\in\mathbb N$ canonical tree-decompositions that distinguish all $k$-distinguishable ends efficiently.

math.CO↗

arXiv2019

A Stallings' type theorem for quasi-transitive graphs

We consider infinite connected quasi-transitive locally finite graphs and show that every such graph with more than one end is a tree amalgamation of two other such graphs. This can be seen as a graph-theoretical version of Stallings' splitting theorem for multi-ended finitely generated groups and indeed it implies this theorem. It will also lead to a characterisation of accessible graphs in terms of tree amalgamations. We obtain applications of our results for hyperbolic graphs, planar graphs and graphs without any thick end. The application for planar graphs answers a question of Mohar in the affirmative.

math.CO↗

arXiv2018

Two-ended quasi-transitive graphs

The well-known characterization of two-ended groups says that every two-ended group can be split over finite subgroups which means it is isomorphic to either by a free product with amalgamation $A\ast_C B$ or an HNN-extension $\ast_ϕ C$, where $C$ is a finite group and $[A:C]=[B:C]=2$ and $ϕ\in Aut(C)$. In this paper, we show that there is a way in order to spilt two-ended quasi-transitive graphs without dominated ends and two-ended transitive graphs over finite subgraphs in the above sense. As an application of it, we characterize all groups acting with finitely many orbits almost freely on those graphs.

math.CO↗

arXiv2017

From cycles to circles in Cayley graphs

For locally finite infinite graphs the notion of Hamilton cycles can be extended to Hamilton circles, homeomorphic images of $S^1$ in the Freudenthal compactification. In this paper we extend some well-known theorems of the Hamiltonicity of finite Cayley graphs to infinite Cayley graphs.

math.CO↗

arXiv2017

Hamilton circles in Cayley graphs

For locally finite infinite graphs the notion of Hamilton cycles can be extended to Hamilton circles, homeomorphic images of $S^1$ in the Freudenthal compactification. In this paper we prove of a sufficient condition for the existence of Hamilton circles in locally finite Cayley graphs.

math.CO↗

arXiv2016

Algebraic flow theory of infinite graphs

A problem by Diestel is to extend algebraic flow theory of finite graphs to infinite graphs with ends. In order to pursue this problem, we define an A-flow and non-elusive H-flow for arbitrary graphs and for abelian topological Hausdorff groups H and compact subsets A of H. We use these new definitions to extend several well-known theorems of flows in finite graphs to infinite graphs.

math.CO↗

arXiv2016

Inverse Limits and Topologies of Infinite Graphs

Two of the natural topologies for infinite graphs with edge-ends are Etop and Itop. In this paper, we study and characterize them. We show that Itop can be constructed by inverse limits of inverse systems of graphs with finitely many vertices. Furthermore, as an application of the inverse limit approach, we construct a topological spanning tree in Itop

math.GN↗

arXiv2015

A Note on Co-Maximal Ideal Graph of Commutative Rings

Let $R$ be a commutative ring with unity. The co-maximal ideal graph of $R$, denoted by $Γ(R)$, is a graph whose vertices are the proper ideals of $R$ which are not contained in the Jacobson radical of $R$, and two vertices $I_1$ and $I_2$ are adjacent if and only if $I_1 + I_2 = R$. We classify all commutative rings whose co-maximal ideal graphs are planar. In 2012 the following question was posed: If $Γ(R)$ is an infinite star graph, can $R$ be isomorphic to the direct product of a field and a local ring? In this paper, we give an affirmative answer to this question.

math.AC↗