id: 06092178
dt: j
an: 06092178
au: Wang, Haiying; Ji, Yang; Li, Chuantao
ti: The sum numbers and the integral sum numbers of the graph $K_{n}\backslash
    E(C_{n-1})$.
so: Ars Comb. 101, 15-26 (2011).
py: 2011
pu: Charles Babbage Research Centre, Winnipeg, MB
la: EN
cc: 
ut: sum graph; integral sum graph; sum number; integral sum number
ci: Zbl 0691.05038; Zbl 0797.05069
li: 
ab: Summary: The concept of the sum graph and integral sum graph were
    introduced by {\it F. Harary} ([Combinatorics, graph theory, and
    computing, Proc. 10th Southeast Conf., Boca Raton/FL (USA) 1989, Congr.
    Numerantium 72, 101‒108 (1990; Zbl 0691.05038)] and [Discrete Math.
    124, No. 1‒3, 99‒105 (1994; Zbl 0797.05069)]). Let $N$ denote the
    set of all positive integers. The sum graph $G^{+}(S)$ of a finite
    subset $S\subset N$ is the graph $(S,E)$ with $uv\in E$ if and only if
    $u+v\in S$. A simple graph $G$ is said to be a sum graph if it is
    isomorphic to a sum graph of some $S\subset N$. The sum number $σ(G)$
    of $G$ is the smallest number of isolated vertices which when added to
    $G$ result in a sum graph. Let $Z$ denote the set of all integers. The
    integral sum graph $G^{+}(S)$ of a finite subset $S\subset Z$ is the
    graph $(S,E)$ with $uv\in E$ if and only $u+v\in S$. A simple graph $G$
    is said to be an integral sum graph if it is isomorphic to an integral
    sum graph of some $S\subset Z$. The integral sum number $ζ(G)$ of $G$
    is the smallest number of isolated vertices which when added to $G$
    result in an integral sum graph. In this paper we investigate and
    determine the sum number and the integral sum number of the graph
    $K_{n}\backslash E(C_{n-1})$. The results are presented as follows: $$
    ζ(K_{n}\backslash E(C_{n-1}))=\cases 0, & n=4,5,6,7\ 2n-7, & n\geq 8 \
    \endcases $$ and $$ σ(K_{n}\backslash E(C_{n-1}))=\cases 1,& n=4\ 2,&
    n=5\ 5,& n=6\ 7,& n=7\ 2n-7,& n\geq 8.\ \endcases $$
rv: