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Let ℕ0 denote the set of all non-negative integers and 풫(ℕ0) be its power set. An integer additive set-indexer (IASI) of a graph G is an injective function f : V(G) → 풫(ℕ0) such that the induced function f+ : E(G) → 풫(ℕ0) defined by f+ (uv) = f(u) + f(v) is also injective. A graph G which admits an IASI is called an integer additive set-indexed graph (IASI-graph). An IASI of a graph G is said to be an arithmetic IASI if the elements of the set-labels of all vertices and edges of G are in arithmetic progressions. In this paper, we discuss about two special types of arithmetic IASIs. [ABSTRACT FROM AUTHOR]

Let G = (V(G), E(G)) be a simple, finite and undirected graph of order n. Given a bijection f: V(G) ∪ E(G) → Zk such that for each edge uv ∈ E(G), f(u) + f(v) + f(uv) is constant C( k). Let nf(i) be the number of vertices and edges labeled by i under f. If |nf(i) - nf(j)| ≤ 1 for all 0 ≤ i < j ≤ k - 1, we say f is a k-totally magic cordial (k-TMC) labeling of G. A graph is said to be k-totally cordial magic if it admits a k-TMC labeling. In this paper, we give some ways to construct new families of k-TMC graphs from a known k-totally cordial magic graphs. We also give a sufficient condition for an odd graph to admit no k-TMC labeling. As a by-product, we determine the k-totally magic cordiality of many families of graphs. [ABSTRACT FROM AUTHOR]