Circulants and their connectivities

There is diverse literature on various properties of a class of graphs known as circulants. We present a new result which answers the previously unsolved question of characterizing the connection sequence of circulants having point connectivity equal to point degree. We also develop some theorems regarding a new generalization of connectivity known as super-connectivity. In addition, we give a survey of published results pertinent to the study of connectivity of circulants.     

On the hamiltonian path graph of a graph

The hamiltonian path graph H(F) of a graph F is that graph having the same vertex set as F and in which two vertices u and v are adjacent if and only if F contains a hamiltonian u ? v path. First, in response to a conjecture of Chartrand, Kapoor and Nordhaus, a characterization of nonhamiltonian graphs isomorphic to their hamiltonian path graphs is presented. Next, the maximum size of a hamiltonian graph F of given order such that K?_d ? H(F) is determined. Finally, it is shown that if the degree sum of the endvertices of a hamiltonian path in a graph F with at least five vertices is at least |V(F)| + t(t ? 0), then H(F) contains a complete subgraph of order t + 4.     

On connectivities of tree graphs

Let T(G) be the tree graph of a graph G with cycle rank r. Then κ(T(G)) ? m(G) ? r, where κ(T(G)) and m(G) denote the connectivity of T(G) and the length of a minimum cycle basis for G, respectively. Moreover, the lower bound of m(G) ? r is best possible.     

The connectivities of adjacent tree graphs

The connectivity and the circuit rank of a graphG are denoted byx(G) and, respectively. It is shown that ifH is the adjacent tree graph of a simple connected graphG, thenx(H)=2.     

On the connectivities of finite and infinite graphs

Results byW. Mader and the authors on the connectivity of finite graphs are generalized to include infinite graphs. In the infinite case a distinction must be made concerning the distribution of finite and infinite components with respect to the separating sets. Results are obtained relating these components to the atoms.     

On the graph for which there is a tree as the inverse interchange graph of a local graph

Let G be a finite, connected, undirected graph without loops and multiple edges. The note modifies slightly the concept of I^–1 (T_t), the inverse interchange graph of the local graph G(T_t) defined by a reference tree t G, and considers the properties of the graph G, when I^–1(T_t) is a tree.     

Notes on the betweenness centrality of a graph

The betweenness centrality of a vertex of a graph is the portion of the shortest paths between all pairs of vertices passing through a given vertex. We study upper bounds for this invariant and its relations to the diameter and average distance of a graph.     

Extending a function on a graph

Let G be a graph with vertex set V, and let h be a function mapping a subset U of V into the real numbers R. If ? is a function from V to R, we define δ (?) to be the sum of ∥?(b)? ?(a)∥ over all edges {a, b} of G. A best extension of h is such a function ? with ?(x) = h(x) for X ∈ U and minimum δ (?). We show that such a best extension exists and derive an algorithm for obtaining such an extension. We also show that if instead we minimise the sum of (?(b)??(a))², there is generally a unique best extension, obtainable by solving a system of linear equations.     

Lower bounds on the cover-index of a graph

In this paper we give some new lower bounds for the cover-index of graphs with multiple edges permitted. The results are analogous to upper bounds for the chromatic index. We show that a simple graph with cover-index different from the minimum degree has at least three vertices of minimum degree. This implies that almost all simple graphs have cover-index equal to the minimum degree.     

Transforming a graph into a 1-balanced graph

Let G be a non-trivial, loopless graph and for each non-trivial subgraph H of G, let . The graph G is 1-balanced if γ(G), the maximum among g(H), taken over all non-trivial subgraphs H of G, is attained when H=G. This quantity γ(G) is called the fractional arboricity of the graph G. The value γ(G) appears in a paper by Picard and Queyranne and has been studied extensively by Catlin, Grossman, Hobbs and Lai. The quantity γ(G)−g(G) measures how much a given graph G differs from being 1-balanced. In this paper, we describe a systematic method of modifying a given graph to obtain a 1-balanced graph on the same number of vertices and edges. We obtain this by a sequence of iterations; each iteration re-defining one end-vertex of an edge in the given graph. After each iteration, either the value γ of the new graph formed is less than that of the graph from the previous iteration or the size of the maximal γ-achieving subgraph of the new graph is smaller than that of the graph in the previous iteration. We show that our algorithm is polynomial in time complexity. Further ways to decrease the number of iterations are also discussed.     

Locating a robber on a graph

Consider the following game of a cop locating a robber on a connected graph. At each turn, the cop chooses a vertex of the graph to probe and receives the distance from the probe to the robber. If she can uniquely locate the robber after this probe, then she wins. Otherwise the robber may either stay put or move to any vertex adjacent to his location other than the probe vertex. The cop’s goal is to minimize the number of probes required to locate the robber, while the robber’s goal is to avoid being located. This is a synthesis of the cop and robber game with the metric dimension problem. We analyse this game for several classes of graphs, including cycles and trees.     

On the dimension to represent a graph by a unit distance graph

It is proved that if a graphG has maximum degreed, then its vertices can be represented by distinct unit vectors inR ^2d so that two vectors are orthogonal if and only if the corresponding vertices are adjacent. As a corollary it follows that if a graph has maximum degreed, then it is isomorphic to a unit distance graph inR ^2d.     

The connectivities of locally finite primitive graphs

Let be an infinite, locally finite graph whose automorphism group is primitive on its vertex set. It is shown that the connectivity of cannot equal 2, but all other values 0, 1, 3, 4, ... are possible.     

Every connected graph is a query graph

Let V be a set of bit strings of length k, i.e., V ? {0, 1}^k. The query graph Q(V) is defined as follows: the vertices of Q(V) are the elements of V, and {ū, v?} is an edge of Q(V) if and only if no other w? ? V agrees with ū in all the positions in which v? does. If V represents the set of keys for a statistical data base in which queries that match only one key are rejected, then the security of the individual data is related to the graph Q(V). Ernst Leiss showed that graphs belonging to any of several classes could be represented as query graphs and asked whether any connected graph could be so represented. We answer his question in the affirmative by making use of a spanning tree with special properties.