For distinct vertices u and v in a graph G, the connectivity between v, denoted $$\kappa _G(u,v)$$ , is maximum number of internally disjoint u–v paths G. The average $${\overline{\kappa }}(G),$$ taken over all unordered pairs u, Analogously, for directed D, from to _D(u,v)$$ D. }}(D)$$ ordered An orientation G obtained by assigning direction every edge let }}_{\max }(G)$$ denote among orientat...

In a 1973 paper, Cooke obtained an upper bound on the possible connectivity of a graph embedded in a surface (orientable or nonorientable) of fixed genus. Furthermore, he claimed that for each orientable genus #>0 (respectively, nonorientable genus # >0, # {2) there is a complete graph of orientable genus # (respectively, nonorientable genus # ) and having connectivity attaining his bound. It i...

We determine the minimum sum–connectivity index of bicyclic graphs with n vertices and matching number m, where 2 ≤ m ≤ ⌊n2 ⌋, the minimum and the second minimum, as well as the maximum and the second maximum sum–connectivity indices of bicyclic graphs with n ≥ 5 vertices. The extremal graphs are characterized. MSC 2000: 05C90; 05C35; 05C07

Measures of network structural integrity useful in the analysis and synthesis of power systems are discussed. Signal flow methodology is applied to derive an expression for the paths between sources and sinks in a power network. Connectivity and reach ability properties of the network are obtained using the minors of a modified connectivity matrix. Node-connectivity, branch connectivity and mix...

Let Γ be a G-symmetric graph whose vertex set admits a nontrivial G-invariant partition B with block size v. Let ΓB be the quotient graph of Γ relative to B and Γ [B,C] the bipartite subgraph of Γ induced by adjacent blocks B,C of B. In this paper we study such graphs for which ΓB is connected, (G,2)-arc transitive and is almost covered by Γ in the sense that Γ [B,C] is a matching of v−1≥2 edge...