The generalized atom-bond connectivity index of a graph G is denoted by ABCa(G) and defined as the sum of weights ((d(u)+d(v)-2)/d(u)d(v))aa$ over all edges uv∊G. A cactus is a graph in which any two cycles have at most one common vertex. In this paper, we compute sharp bounds for  ABCa index for cacti of order $n$ ...

For any integer $k\geq 3$ , we define sunlet graph of order $2k$, denoted by $L_{2k}$, as the consisting a cycle length $k$ together with pendant vertices, each adjacent to exactly one vertex cycle. In this paper, give necessary and sufficient conditions for existence $L_{8}$-decomposition tensor product wreath complete graphs.

Here we denote a diameter six tree by (c; a1, a2, . . . , am; b1, b2, . . . , bn; c1, c2, . . . , cr), where c is the center of the tree; ai, i = 1, 2, . . . ,m, bj, j = 1, 2, . . . , n, and ck, k = 1, 2, . . . , r are the vertices of the tree adjacent to c; each ai is the center of a diameter four tree, each bj is the center of a star, and each ck is a pendant vertex. Here we give graceful lab...