The extremal function for 3-linked graphs

The extremal function for 3-linked graphs

A graph is k-linked if for every set of 2k distinct vertices {s1, . . . , sk, t1, . . . , tk} there exist disjoint paths P1, . . . , Pk such that the endpoints of Pi are si and ti. We prove every 6-connected graph on n vertices with 5n − 14 edges is 3-linked. This is optimal, in that there exist 6-connected graphs on n vertices with 5n− 15 edges that are not 3-linked for arbitrarily large value...

An extremal problem for H-linked graphs

We introduce the notion of H-linked graphs, where H is a fixed multigraph with vertices w1; . . . ;wm. A graph G is H-linked if for every choice of vertices v1; . . . ; vm in G, there exists a subdivision of H in G such that vi is the branch vertex representing wi (for all i). This generalizes the notions of k -linked, k -connected, and k-ordered graphs. Given k and n 5k þ 6, we determine the l...

Eccentric Connectivity Index: Extremal Graphs and Values

Eccentric connectivity index has been found to have a low degeneracy and hence a significant potential of predicting biological activity of certain classes of chemical compounds. We present here explicit formulas for eccentric connectivity index of various families of graphs. We also show that the eccentric connectivity index grows at most polynomially with the number of vertices and determine ...

The Extremal Graphs for (Sum-) Balaban Index of Spiro and Polyphenyl Hexagonal Chains

As highly discriminant distance-based topological indices, the Balaban index and the sum-Balaban index of a graph $G$ are defined as $J(G)=frac{m}{mu+1}sumlimits_{uvin E} frac{1}{sqrt{D_{G}(u)D_{G}(v)}}$ and $SJ(G)=frac{m}{mu+1}sumlimits_{uvin E} frac{1}{sqrt{D_{G}(u)+D_{G}(v)}}$, respectively, where $D_{G}(u)=sumlimits_{vin V}d(u,v)$ is the distance sum of vertex $u$ in $G$, $m$ is the n...

Extremal graphs for weights