The Wiener Index Of K Polygons Cactus Graphs

This paper mainly investigates the Wiener indices of tree-like k edges Cactus graphs,firstly,the properties and extremal graphs of Wiener indices of tree-like k edges Cactus graphs areobtained, secondly, it gives the exact upper bound and lower bound of Wiener indices in tree-like k edges Cactus graphs with the length of n, the last, the extremum of k edges Cactuschain are also given.In the followings, the main results are presented:定理2.1 let Y,Y~+∈C(h),then W(Y )≡W(Y~+)mod(k-1).定理3.1 let G∈C(h),thenwhen h≥1,W(Fh)≤W(Th);whenh≥3,W(Fh) < W(Th).定理3.2 let G∈C(h),thenwhen h≥1,W(G)≤W(Lh);when h≥3,W(G) < W(Lh).定理3.3 let G∈C(h),thenwhen h≥1,W(Fh)≤W(G)≤W(Lh);when h≥3,W(Fh) < W(G) < W(Lh).定理4.1 let G∈C(h) be a chain Cactus graph, thenW(H(h))≤W(G)≤W(L(h))further W(H(h)) = W(G) if and only if H(h) = G;W(G) = W(L(h)) if and only ifG = L(h).