(a,d)-Continuous monotonic subgraph decomposition of K_n+1 and integral sum graphs G_(0,n)

Abstract

For $a,d,n \in \mathbb{N}$, we define$(a, d)-Continuous$ Monotonic Subgraph Decomposition or $(a,d)-CMSD$ of a graph $G$ of size $\frac{(2a+(n-1)d)n}{2}$ as the decomposition of $G$ into $n$ subgraphs $G_1,G_2,\ldots,G_n$ without isolated vertices such that each $G_i$ is connected and isomorphic to a proper subgraph of $G_{i+1}$ and $|E(G_i)| = a+(i-1)d$ for $i = 1,2,\ldots,n.$ $(1, 1)-CMSD$ of a graph $G$ is called a Continuous Monotonic Subgraph Decomposition or CMSD of $G$. Harary introduced the concepts of sum and integral sum graphs and a family of integral sum graphs $G_{-n,n}$ over $[-n, n]$ and it was generalized to $G_{-m,n}$ where $[r, s] = \{r,r+1,\ldots,s\}$, $r,s \in \mathbb{Z}$ and $m,n \in \mathbb{N}_0$. In this paper, we study $(a, d)-CMSD$ of $K_{n+1}$ and $G_{0,n}$ into families of triangular books, triangular books with book mark and Fans with handle.