Connectivity of vertex and edge transitive graphs

A graph is said to be super-connected if every minimum vertex cut isolates a vertex. A graph is said to be hyper-connected if the deletion of each minimum vertex cut creates exactly two components, one of which is an isolated vertex. In this note, we proved that a vertex transitive bipartite graph is not super-connected if and only if it is isomorphic to the lexicographic product of a cycle Cn(n ≥ 6) by a null graph Nm. We also characterized non-hyper-connected vertex transitive bipartite graphs.

Semi-hyper-connected edge transitive graphs

In the recent paper, authors introduced a graph topological structure, called open subset intersection graph of a topological space ϒ(τ) on a finite set X. In this present paper, we continue the study of the graph ϒ(τ) on a finite set X. It is shown that, if τ is a discrete topology on X and |X| ≥ 3, then the graph ϒ(τ) is not bipartite and regular. If τ is a discrete topology on X with |X| = 3 then it is shown that the ϒ(τ) is an edge-transitive graph, Perfect graph, Eulerian and Hamiltonian graph . Moreover, we detremine exact value of the independence number, vertex connectivity and edge connectivity of the graph ϒ(τ).

Inequalities for the spectra of symmetric doubly stochastic matrices

It is known that the Levi graph of any rank two coset geometry is an edge-transitive graph, and thus coset geometries can be used to construct many edge transitive graphs. In this paper, we consider the reverse direction. Starting from edge-transitive graphs, we construct all associated core-free, rank two coset geometries. In particular, we focus on 3-valent and 4-valent graphs, and are able to construct coset geometries arising from these graphs. We summarize many properties of these coset geometries in a sequence of tables; in the 4-valent case we restrict to graphs that have relatively small vertex-stabilizers.

The cyclic vertex (resp., edge) connectivity of a graph [Formula: see text], denoted by [Formula: see text] (resp., [Formula: see text]), is the minimum number of vertices (resp., edges) whose removal from [Formula: see text] results in a disconnected graph and at least two remaining components contain cycles. Thus, to determine the exact values of [Formula: see text] and [Formula: see text] is important in the reliability assessment of interconnection networks. However, the study of the cyclic vertex (edge) connectivity is less involved. In this paper, we determine the cyclic vertex (edge) connectivity of the burnt pancake graphs [Formula: see text] which is the Cayley graph of the group of signed permutations using prefix reversals as generators. By exploring the combinatorial properties and fault-tolerance of [Formula: see text], we show [Formula: see text] and [Formula: see text] for [Formula: see text]. Moreover, we determine that [Formula: see text] for [Formula: see text].

A note on eigenvalue, spectral radius and energy of extended adjacency matrix

The research of my dissertation was motivated by the conjecture of Thomassen that every 4-connected line graph is hamiltonian and by the conjecture of Matthews and Sumner that every 4-connected claw-free graph is hamiltonian. Towards the hamiltonian line graph problem, we proved that every 3-edge-connected, essentially 4-edge-connected graph G has a spanning eulerian subgraph, if for every pair of adjacent vertices u and v, dG(u) + dG(v) ≥ 9. A straight forward corollary is that every 4-connected, essentially 6-connected line graph with minimum degree at least 7 is hamiltonian.;We also investigate graphs G such that the line graph L(G) is hamiltonian connected when L( G) is 4-connected. Ryjacek and Vrana recently further conjectured that every 4-connected line graph is hamiltonian-connected. In 2001, Kriesell proved that every 4-connected line graph of a claw free graph is hamiltonian connected. Recently, Lai et al showed that every 4-connected line graph of a quasi claw free graph is hamiltonian connected, and that every 4-connected line graph of an almost claw free graph is hamiltonian connected. In 2009, Broersma and Vumer discovered the P3-dominating (P3D) graphs as a superfamily that properly contains all quasi claw free graphs, and in particular, all claw-free graphs. Here we prove that every 4-connected line graph of a P3D graph is hamiltonian connected, which extends several former results in this area.;R. Gould [15] asked what natural graph properties of G and H are sufficient to imply that the product of G and H is hamiltonian. We first investigate the sufficient and necessary conditions for G x H being hamiltonian or traceable when G is a hamiltonian graph and H is a tree. Then we further investigate sufficient and necessary conditions for G x H being hamiltonian connected, or edge-pancyclic, or pan-connected.;The problem of group colorings of graphs is also investigated in this dissertation. Group coloring was first introduced by Jeager et al. [21]. They introduced a concept of group connectivity as a generalization of nowhere-zero flows. They also introduced group coloring as a dual concept to group connectivity. Prior research on group chromatic number was restricted to simple graphs, and considered only Abelian groups in the definition of chi g(G). The behavior of group coloring for multigraphs is different to that of simple graphs. Thus we extend the definition of group coloring by considering general groups (both Abelian groups and non-Abelian groups), and investigate the properties of chig for multigraphs

The vertex connectivity of an m-edge n-vertex undirected graph is the smallest number of vertices whose removal disconnects the graph, or leaves only a singleton vertex. In this paper, we give a reduction from the vertex connectivity problem to a set of maxflow instances. Using this reduction, we can solve vertex connectivity in (mα) time for any α ≥ 1, if there is a mα-time maxflow algorithm. Using the current best maxflow algorithm that runs in m4/3+o(1) time (Kathuria, Liu and Sidford, FOCS 2020), this yields a m4/3+o(1)-time vertex connectivity algorithm. This is the first improvement in the running time of the vertex connectivity problem in over 20 years, the previous best being an O(mn)-time algorithm due to Henzinger, Rao, and Gabow (FOCS 1996). Indeed, no algorithm with an o(mn) running time was known before our work, even if we assume an (m)-time maxflow algorithm. Our new technique is robust enough to also improve the best O(mn)-time bound for directed vertex connectivity to mn1−1/12+o(1) time

The decomposition of a connected graph by the set of its cut-vertices, sometimes called the \\"block decomposition\\" or \\"block tree\\" of a graph, is a well known and basic concept in graph theory. This decomposition, however, does not provide meaningful information when applied to a k-connected graph for k > 1. There has been a number of attempts to generalize the construction of the block decomposition of a graph for the case of k-connected graphs. Notably, Tutte constructed a tree that describes the mutual arrangement of 2-cutsets in a 2-connected graph. This decomposition has some similarities to the block decomposition of a connected graph. In other works, a block of a k-connected graph was defined as a maximal (k+1)-connected subgraph. Karpov described the decomposition of a k-connected graph by the set of k-cutsets that are not separated by any other k-cutset of the graph. Karpov also described some special properties of his decomposition for the case of a 2-connected graph. The decompositions defined by Karpov and Tutte for the case of a 2-connected graph share some similarities. In this work, we present a self-contained description of Karpov\\'s decomposition. We also present some applications to the study of planarity, the chromatic number, critically 2-connected graphs, and the partition of certain 2-connected graphs into three connected subgraphs.

On super restricted edge-connectivity of edge-transitive graphs

Edge and vertex connectivity, as well as edge and vertex cuts are among the most basic and fundamental concepts in graph theory. In particular, they are naturally significant in a networking context as they are a measure for the rate at which information can be transferred across a network. While in a traditional, sequential setting, there is a rich literature (in particular on problems related to edge connectivity and edge cuts), until recently, much less was known from a distributed algorithms point of view. In my talk, I will discuss and give partial answers to some of the following basic questions. Using distributed algorithms, how fast can we compute or approximate the edge or vertex connectivity and is it possible to efficiently find small cuts in a network? Assuming, we have a network with good connectivity properties, to what extent is it possible to exploit this in order to speed up distributed computations? Where such properties can be exploited, are there network structures that allow to make use of good connectivity in a structured (and somewhat canonical) way and can we construct such structures in a distributed manner?

We investigate the generalized line graph, total graph and middle graph of the given graphs. First, replace a given graph G with a multigraph M having the same point set as G, then it constructs the total graph T(G) from G. Two suitable subgraphs Mx(G), Lx(G), of Tx(G) are considered as extensions of middle graph M(G) and line graph L(G) respectively.We also consider the connectivities of graphs which are obtained as a special case of Lx(G), Mx(G) and Tx(G). Then we get some extensions of established theorems.

A graph $G = (C, I, E)$ is called a split graph if its vertex set $V$ can be partitioned into a clique $C$ and an independent set $I$. A graph $G = (Y \cup Z, I, E)$ is called a bisplit graph if its vertex set $V$ can be partitioned into three stable sets $I, Y,Z$ such that $Y \cup Z$ induces a complete bipartite graph and an independent set $I$. A connected graph $G$ is called supper-$κ$ (resp. super-$λ$) if every minimum vertex cut (edge cut) of $G$ is the set of neighbors of some vertex (the edges of incident to some vertex) in $G$. In this note, we show that: split graphs and bisplit graphs are super-$κ$ and super-$λ$.

It is shown that the vertex connectivity of the block-intersection graph of a balanced incomplete block design,BIBD (v, k, 1), is equal to its minimum degree. A similar statement is proved for the edge connectivity of the block-intersection graph of a pairwise balanced design,PBD (v, K, 1). A partial result on the vertex connectivity of these graphs is also given. Minimal vertex and edge cuts for the corresponding graphs are characterized.