Further topics in connectivity

Continuing the study of connectivity, initiated in §4.1 of the Handbook, we survey here some (sufficient) conditions under which a graph or digraph has a given connectivity or edge-connectivity. First, we describe results concerning maximal (vertex- or edge-) connectivity. Next, we deal with conditions for having (usually lower) bounds for the connectivity parameters. Finally, some other general connectivity measures, such as one instance of the so-called “conditional connectivity,” are considered. For unexplained terminology concerning connectivity, see §4.1.Peer ReviewedPostprint (published version

On the structure of graphs without short cycles

The objective of this thesis is to study cages, constructions and properties of such families of graphs. For this, the study of graphs without short cycles plays a fundamental role in order to develop some knowledge on their structure, so we can later deal with the problems on cages. Cages were introduced by Tutte in 1947. In 1963, Erdös and Sachs proved that (k, g) -cages exist for any given values of k and g. Since then, large amount of research in cages has been devoted to their construction. In this work we study structural properties such as the connectivity, diameter, and degree regularity of graphs without short cycles. In some sense, connectivity is a measure of the reliability of a network. Two graphs with the same edge-connectivity, may be considered to have different reliabilities, as a more refined index than the edge-connectivity, edge-superconnectivity is proposed together with some other parameters called restricted connectivities. By relaxing the conditions that are imposed for the graphs to be cages, we can achieve more refined connectivity properties on these families and also we have an approach to structural properties of the family of graphs with more restrictions (i.e., the cages). Our aim, by studying such structural properties of cages is to get a deeper insight into their structure so we can attack the problem of their construction. By way of example, we studied a condition on the diameter in relation to the girth pair of a graph, and as a corollary we obtained a result guaranteeing restricted connectivity of a special family of graphs arising from geometry, such as polarity graphs. Also, we obtained a result proving the edge superconnectivity of semiregular cages. Based on these studies it was possible to develop the study of cages. Therefore obtaining a relevant result with respect to the connectivity of cages, that is, cages are k/2-connected. And also arising from the previous work on girth pairs we obtained constructions for girth pair cages that proves a bound conjectured by Harary and Kovács, relating the order of girth pair cages with the one for cages. Concerning the degree and the diameter, there is the concept of a Moore graph, it was introduced by Hoffman and Singleton after Edward F. Moore, who posed the question of describing and classifying these graphs. As well as having the maximum possible number of vertices for a given combination of degree and diameter, Moore graphs have the minimum possible number of vertices for a regular graph with given degree and girth. That is, any Moore graph is a cage. The formula for the number of vertices in a Moore graph can be generalized to allow a definition of Moore graphs with even girth (bipartite Moore graphs) as well as odd girth, and again these graphs are cages. Thus, Moore graphs give a lower bound for the order of cages, but they are known to exist only for very specific values of k, therefore it is interesting to study how far a cage is from this bound, this value is called the excess of a cage. We studied the excess of graphs and give a contribution, in the sense of the work of Biggs and Ito, relating the bipartition of girth 6 cages with their orders. Entire families of cages can be obtained from finite geometries, for example, the graphs of incidence of projective planes of order q a prime power, are (q+1, 6)-cages. Also by using other incidence structures such as the generalized quadrangles or generalized hexagons, it can be obtained families of cages of girths 8 and 12. In this thesis, we present a construction of an entire family of girth 7 cages that arises from some combinatorial properties of the incidence graphs of generalized quadrangles of order (q,q)

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

A 3−arc of a graph G is a 4-tuple (y, a, b, x) of vertices such that both (y, a, b) and (a, b, x) are paths of length two in G. Let ←→G denote the symmetric digraph of a graph G. The 3-arc graph X(G) of a given graph G is defined to have vertices the arcs of ←→G. Two vertices (ay), (bx) are adjacent in X(G) if and only if (y, a, b, x) is a 3-arc of G. The purpose of this work is to study the edge-connectivity and restricted edge-connectivity of 3-arc graphs.We prove that the 3-arc graph X(G) of every connected graph G of minimum degree δ(G) ≥ 3 has edge-connectivity λ(X(G)) ≥ (δ(G) − 1)2; and restricted edge- connectivity λ(2)(X(G)) ≥ 2(δ(G) − 1)2 − 2 if κ(G) ≥ 2. We also provide examples showing that all these bounds are sharp.Peer Reviewe

Superconnectivity of Networks Modeled by the Strong Product of Graphs

Maximal connectivity and superconnectivity in a network are two important features of its reliability. In this paper, using graph terminology, we first give a lower bound for the vertex connectivity of the strong product of two networks and then we prove that the resulting structure is more reliable than its generators. Namely, sufficient conditions for a strong product of two networks to be maximally connected and superconnected are given.Ministerio de Economía y Competitividad MTM2014-60127-

Proceedings of the 3rd International Workshop on Optimal Networks Topologies IWONT 2010

Peer Reviewe

Extra Connectivity of Strong Product of Graphs

The $g$-$extra$ $connectivity$ $\kappa_{g}(G)$ of a connected graph $G$ is the minimum cardinality of a set of vertices, if it exists, whose deletion makes $G$ disconnected and leaves each remaining component with more than $g$ vertices, where $g$ is a non-negative integer. The $strong$ $product$ $G_1 \boxtimes G_2$ of graphs $G_1$ and $G_2$ is the graph with vertex set $V(G_1 \boxtimes G_2)=V(G_1)\times V(G_2)$, where two distinct vertices $(x_{1}, y_{1}),(x_{2}, y_{2}) \in V(G_1)\times V(G_2)$ are adjacent in $G_1 \boxtimes G_2$ if and only if $x_{1}=x_{2}$ and $y_{1} y_{2} \in E(G_2)$ or $y_{1}=y_{2}$ and $x_{1} x_{2} \in E(G_1)$ or $x_{1} x_{2} \in E(G_1)$ and $y_{1} y_{2} \in E(G_2)$. In this paper, we give the $g\ (\leq 3)$-$extra$ $connectivity$ of $G_1\boxtimes G_2$, where $G_i$ is a maximally connected $k_i\ (\geq 2)$-regular graph for $i=1,2$. As a byproduct, we get $g\ (\leq 3)$-$extra$ conditional fault-diagnosability of $G_1\boxtimes G_2$ under $PMC$ model

Sufficient conditions for super k-restricted edge connectivity in graphs of diameter 2

AbstractFor a connected graph G=(V,E), an edge set S⊆E is a k-restricted edge cut if G−S is disconnected and every component of G−S has at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is defined as the cardinality of a minimum k-restricted edge cut. Let ξk(G)=min{|[X,X¯]|:|X|=k,G[X]is connected}. G is λk-optimal if λk(G)=ξk(G). Moreover, G is super-λk if every minimum k-restricted edge cut of G isolates one connected subgraph of order k. In this paper, we prove that if |NG(u)∩NG(v)|≥2k−1 for all pairs u, v of nonadjacent vertices, then G is λk-optimal; and if |NG(u)∩NG(v)|≥2k for all pairs u, v of nonadjacent vertices, then G is either super-λk or in a special class of graphs. In addition, for k-isoperimetric edge connectivity, which is closely related with the concept of k-restricted edge connectivity, we show similar results

Avances en Matemática Discreta en Andalucía. V Encuentro andaluz de Matemática Discreta. La Línea de la Concepción (Cádiz), 4-5 de julio de 2007

V Encuentro andaluz de Matemática Discreta. La Línea de la Concepción (Cádiz), 4-5 de julio de 200

Extremal Directed And Mixed Graphs

We consider three problems in extremal graph theory, namely the degree/diameter problem, the degree/geodecity problem and Tur\'{a}n problems, in the context of directed and partially directed graphs. A directed graph or mixed graph $G$ is $k$-geodetic if there is no pair of vertices $u,v$ of $G$ such that there exist distinct non-backtracking walks with length $\leq k$ in $G$ from $u$ to $v$. The order of a $k$-geodetic digraph with minimum out-degree $d$ is bounded below by the \emph{directed Moore bound} $M(d,k) = 1+d+d^2+\dots +d^k$; similarly the order of a $k$-geodetic mixed graph with minimum undirected degree $r$ and minimum directed out-degree $z$ is bounded below by the \emph{mixed Moore bound}. We will be interested in networks with order exceeding the Moore bound by some small \emph{excess} $\epsilon$. The \emph{degree/geodecity problem} asks for the smallest possible order of a $k$-geodetic digraph or mixed graph with given degree parameters. We prove the existence of extremal graphs, which we call \emph{geodetic cages}, and provide some bounds on their order and information on their structure. We discuss the structure of digraphs with excess one and rule out the existence of certain digraphs with excess one. We then classify all digraphs with out-degree two and excess two, as well as all diregular digraphs with out-degree two and excess three. We also present the first known non-trivial examples of geodetic cages. We then generalise this work to the setting of mixed graphs. First we address the question of the total regularity of mixed graphs with order close to the Moore bound and prove bounds on the order of mixed graphs that are not totally regular. In particular using spectral methods we prove a conjecture of L\'{o}pez and Miret that mixed graphs with diameter two and order one less than the Moore bound are totally regular. Using counting arguments we then provide strong bounds on the order of totally regular $k$-geodetic mixed graphs and use these results to derive new extremal mixed graphs. Finally we change our focus and study the Tur\'{a}n problem of the largest possible size of a $k$-geodetic digraph with given order. We solve this problem and also prove an exact expression for the restricted problem of the largest possible size of strongly connected $2$-geodetic digraphs, as well as providing constructions of strongly connected $k$-geodetic digraphs that we conjecture to be extremal for larger $k$. We close with a discussion of some related generalised Tur\'{a}n problems for $k$-geodetic digraphs