11,871 research outputs found

    The generalized 3-connectivity of Cartesian product graphs

    Full text link
    The generalized connectivity of a graph, which was introduced recently by Chartrand et al., is a generalization of the concept of vertex connectivity. Let SS be a nonempty set of vertices of GG, a collection {T1,T2,...,Tr}\{T_1,T_2,...,T_r\} of trees in GG is said to be internally disjoint trees connecting SS if E(Ti)∩E(Tj)=∅E(T_i)\cap E(T_j)=\emptyset and V(Ti)∩V(Tj)=SV(T_i)\cap V(T_j)=S for any pair of distinct integers i,ji,j, where 1≤i,j≤r1\leq i,j\leq r. For an integer kk with 2≤k≤n2\leq k\leq n, the kk-connectivity κk(G)\kappa_k(G) of GG is the greatest positive integer rr for which GG contains at least rr internally disjoint trees connecting SS for any set SS of kk vertices of GG. Obviously, κ2(G)=κ(G)\kappa_2(G)=\kappa(G) is the connectivity of GG. Sabidussi showed that κ(G□H)≥κ(G)+κ(H)\kappa(G\Box H) \geq \kappa(G)+\kappa(H) for any two connected graphs GG and HH. In this paper, we first study the 3-connectivity of the Cartesian product of a graph GG and a tree TT, and show that (i)(i) if κ3(G)=κ(G)≥1\kappa_3(G)=\kappa(G)\geq 1, then κ3(G□T)≥κ3(G)\kappa_3(G\Box T)\geq \kappa_3(G); (ii)(ii) if 1≤κ3(G)<κ(G)1\leq \kappa_3(G)< \kappa(G), then κ3(G□T)≥κ3(G)+1\kappa_3(G\Box T)\geq \kappa_3(G)+1. Furthermore, for any two connected graphs GG and HH with κ3(G)≥κ3(H)\kappa_3(G)\geq\kappa_3(H), if κ(G)>κ3(G)\kappa(G)>\kappa_3(G), then κ3(G□H)≥κ3(G)+κ3(H)\kappa_3(G\Box H)\geq \kappa_3(G)+\kappa_3(H); if κ(G)=κ3(G)\kappa(G)=\kappa_3(G), then κ3(G□H)≥κ3(G)+κ3(H)−1\kappa_3(G\Box H)\geq \kappa_3(G)+\kappa_3(H)-1. Our result could be seen as a generalization of Sabidussi's result. Moreover, all the bounds are sharp.Comment: 17 page

    The generalized 3-connectivity of Lexicographic product graphs

    Full text link
    The generalized kk-connectivity κk(G)\kappa_k(G) of a graph GG, introduced by Chartrand et al., is a natural and nice generalization of the concept of (vertex-)connectivity. In this paper, we prove that for any two connected graphs GG and HH, κ3(G∘H)≥κ3(G)∣V(H)∣\kappa_3(G\circ H)\geq \kappa_3(G)|V(H)|. We also give upper bounds for κ3(G□H)\kappa_3(G\Box H) and κ3(G∘H)\kappa_3(G\circ H). Moreover, all the bounds are sharp.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1103.609

    Computing Reformulated First Zagreb Index of Some Chemical Graphs as an Application of Generalized Hierarchical Product of Graphs

    Full text link
    The generalized hierarchical product of graphs was introduced by L. Barri\'ere et al in 2009. In this paper, reformulated first Zagreb index of generalized hierarchical product of two connected graphs and hence as a special case cluster product of graphs are obtained. Finally using the derived results the reformulated first Zagreb index of some chemically important graphs such as square comb lattice, hexagonal chain, molecular graph of truncated cube, dimer fullerene, zig-zag polyhex nanotube and dicentric dendrimers are computed.Comment: 12 page

    The generalized 3-edge-connectivity of lexicographic product graphs

    Full text link
    The generalized kk-edge-connectivity λk(G)\lambda_k(G) of a graph GG is a generalization of the concept of edge-connectivity. The lexicographic product of two graphs GG and HH, denoted by G∘HG\circ H, is an important graph product. In this paper, we mainly study the generalized 3-edge-connectivity of G∘HG \circ H, and get upper and lower bounds of λ3(G∘H)\lambda_3(G \circ H). Moreover, all bounds are sharp.Comment: 14 page
    • …
    corecore