Neighborhoods, connectivity, and diameter of the nilpotent graph of a finite group

Decomposition of 3-connected cubic graphs

Fault diameter is an important parameter to measure the reliability and efficiency of interconnection networks. Strong product is an efficient method to construct large graphs from small graphs. In this paper, we study the fault diameter of strong product graph of [Formula: see text] paths. By recursion and mathematical induction, combined with the connectivity of the strong product graph of complete graph and connected graph, the connectivity of the strong product graph of [Formula: see text] paths is determined. By defining two [Formula: see text]-dimensional vectors [Formula: see text] and [Formula: see text] to construct internally vertex disjoint paths between any two vertices, the fault diameter of the strong product graph of [Formula: see text] paths is determined.

Let Gbe a finite p-solvable group. Let us consider the graph Γ* p (G) whose vertices are the primes which occur as the divisors of the conjugacy classes of p-regular elements of G and two primes are joined by an edge if there exists such a class whose size is divisible by both primes. Suppose that Γ p *(G) is a connected graph, then we prove that the diameter of this graph is at most 3 and this is the best bound.

Diameter and connectivity of 3-arc graphs

The maximum genus of graphs of diameter two

In this thesis a heuristic method for factoring semiprimes by multiagent depth-limited search of PG 2 N graphs is presented. An analysis of PG 2 N graph connectivity is used to generate heuristics for multiagent search. Further analysis is presented including the requirements on choosing prime numbers to generate ’hard’ semiprimes; the lack of connectivity in PG 1 N graphs; the counts of spanning trees inPG 2 N graphs; the upper boundof aPG 2 N graph diameter and a conjecture on the frequency distribution of prime numbers on Hamming distance. We further demonstrated the feasibility of the HD2 breadth first search of PG 2 N graphs for factoring small semiprimes. We presented the performance of different multiagent search heuristics in PG 2 N graphs showing that the heuristic of most connected seedpick outperforms least connected or random connected seedpick heuristics on small PG 2 N graphs of sizeN ≤ 26. The contribution of this small scale research was to develop heuristics for seed selection that may extrapolate to larger values ofN .

For \documentclass{article} \usepackage{amsmath,amsfonts,amssymb}\pagestyle{empty}\begin{document} $n\in \mathbb{N}$ \end{document} and \documentclass{article} \usepackage{amsmath,amsfonts,amssymb}\pagestyle{empty}\begin{document} $D\subseteq \mathbb{N}$ \end{document}, the distance graph P has vertex set {0,1,…,n − 1} and edge set {ij | 0 ≤ i,j ≤ n − 1,|j − i| ∈ D}. The class of distance graphs generalizes the important and very well-studied class of circulant graphs, which have been proposed for numerous network applications. In view of fault tolerance and delay issues in these applications, the connectivity and diameter of circulant graphs have been studied in great detail. Our contributions are hardness results concerning computational problems related to the connectivity and the diameter of distance graphs and a characterization of the connected distance graphs P for |D| = 2. © 2010 Wiley Periodicals, Inc. NETWORKS, Vol. 57(4), 310-315 2011

On total domination vertex critical graphs of high connectivity

Let κ(G) denote the (vertex) connectivity of a graph G. For l≥0, a noncomplete graph of finite connectivity is called l-critical if κ(G−X)=κ(G)−|X| for every X⊆V(G) with |X|≤l.Mader proved that every 3-critical graph has diameter at most 4 and asked for 3-critical graphs having diameter exceeding 2. Here we give an affirmative answer by constructing an l-critical graph of diameter 3 for every l≥3.

On the connectivity and restricted edge-connectivity of 3-arc graphs

The visibility graph of a finite set of points in the plane has the points as vertices and an edge between two vertices if the line segment between them contains no other points. This paper establishes bounds on the edge- and vertex-connectivity of visibility graphs. Unless all its vertices are collinear, a visibility graph has diameter at most 2, and so it follows by a result of Plesnik (Acta Fac. Rerum Nat. Univ. Comen. Math. 30:71–93, 1975) that its edge-connectivity equals its minimum degree. We strengthen the result of Plesnik by showing that for any two vertices v and w in a graph of diameter 2, if deg(v)≤deg(w) then there exist deg(v) edge-disjoint vw-paths of length at most 4. For vertex-connectivity, we prove that every visibility graph with n vertices and at most l collinear vertices has connectivity at least $\frac{n-1}{\ell-1}$, which is tight. We also prove the qualitatively stronger result that the vertex-connectivity is at least half the minimum degree. Finally, in the case that l=4 we improve this bound to two thirds of the minimum degree.

Spherical fullerene graphs that do not satisfy Andova and Škrekovski's conjecture

On the connectivity and the diameter of betweenness-uniform graphs

The distinguishing number of a group $G$ acting faithfully on a set $V$ is the least number of colors needed to color the elements of $V$ so that no non-identity element of the group preserves the coloring. The distinguishing number of a graph is the distinguishing number of its automorphism group acting on its vertex set. A connected graph $\Gamma$ is said to have connectivity 1 if there exists a vertex $\alpha \in V\Gamma$ such that $\Gamma \setminus \{\alpha\}$ is not connected. For $\alpha \in V$, an orbit of the point stabilizer $G_\alpha$ is called a suborbit of $G$.We prove that every nonnull, primitive graph with infinite diameter and countably many vertices has distinguishing number $2$. Consequently, any nonnull, infinite, primitive, locally finite graph is $2$-distinguishable; so, too, is any infinite primitive permutation group with finite suborbits. We also show that all denumerable vertex-transitive graphs of connectivity 1 and all Cartesian products of connected denumerable graphs of infinite diameter have distinguishing number $2$. All of our results follow directly from a versatile lemma which we call The Distinct Spheres Lemma.

In 2011, Caro et al. introduced the monochromatic connection of graphs. An edge-coloring of a connected graph $G$ is called a monochromatically connecting (MC-coloring, for short) if there is a monochromatic path joining any two vertices. The monochromatic connection number $\operatorname{mc}(G)$ of a graph $G$ is the maximum integer $k$ such that there is a $k$-edge-coloring, which is an MC-coloring of $G$. Clearly, a monochromatic spanning tree can monochromatically connect any two vertices. So for a graph $G$ of order $n$ and size $m$, $\operatorname{mc}(G) \geq m-n+2$. Caro et al. proved that both triangle-free graphs and graphs of diameter at least three meet the lower bound. In this paper, we consider the monochromatic connectivity of graphs containing triangles which meet the lower bound too. Also, in order to study the graphs of diameter two, we present the formula for the monochromatic connectivity of join graphs. This will be helpful to solve the problem for graphs of diameter two.