Abstract: A connected graph G = (V,E) of order atleast two, with order p and size q is called vertex-graceful if there exists a bijection ʄ : V → { 1, 2, 3, ··· p } such that the induced function ʄ*: E → { 0, 1, 2, ··· q-1} defined by ʄ*(uv) = (ʄ(u)+ ʄ(v)) (mod q) is a bijection. The bijection ʄ is called a vertex-graceful labeling of G. A subset S of the set of natural numbers N is called consecutive if S consists of consecutive integers. For any set X, a mapping ʄ : X → N $ is said to be consecutive if ʄ(X) is consecutive. A vertex-graceful labeling ʄ is said to be strong if the function ʄ1: E → N defined by ʄ1(e)= ʄ(u) + ʄ(v) for all edges e = uv in E forms a consecutive set. It is proved that one vertex union of odd number of copies of isomorphic caterpillars is vertex-graceful and any caterpillar is strong vertex-graceful. It is proved that a spider with even number of legs (paths) of equal length appended to each vertex of an odd cycle is vertex-graceful. It is also proved that the graph lA(mj,n) is vertex-graceful for both n and l odd, 0 ≤ i ≤ n-1, 1 ≤ j ≤ mi. Further, it is proved that the graph A(mj, n) is strong vertex-graceful for n odd, 0 ≤ i ≤ n-1, 1 ≤ j ≤ mi.

Abstract In this study, the glass/epoxy composite laminate is layered with polyurethane foam/ polyurethane sheet and silicon carbide to analyse their response during high mass and low velocity impact. The silicon carbide is layered in two forms, one is as plate and the other is as inserts. The target materials are prepared in various combinations and the bonding of layers is done by using epoxy. Effectiveness of silicon carbide inserts and plates are compared in terms of their energy absorbing capacities. The numerical simulation is also carried for the target material with the same experimental conditions. The experimental results are compared with the numerical results for validation and a reasonably good agreement is found. Further, the validated numerical model is extended to understand the ballistic performance of the target material. It is observed that the introduction of silicon carbide as front layer improves both the structural and ballistic performance. Also, the damage in case of samples with silicon carbide inserts is localized as opposed to that of silicon carbide plate.