Integral Sum Number Research Articles

Some results on the spum and the integral spum of graphs

A finite simple graph G is called a sum graph (integral sum graph) if there is a bijectionf from the vertices of G to a set of positive integers S (a set of integers S) such that uv is an edge of G if and only if f(u)+f(v)∈S. For a connected graph G, the sum number (the integral sum number) of G, denoted by σ(G) (ζ(G)), is the minimum number of isolated vertices that must be added to G so that the resulting graph is a sum graph (an integral sum graph). The spum (the integral spum) of a graph G is the minimum difference between the largest and smallest integer in any set S that corresponds to a sum graph (integral sum graph) containing G. We investigate the spum and integral spum of several classes of graphs, including complete graphs, symmetric complete bipartite graphs, star graphs, cycles, and paths. We also give sharp lower bounds for the spum and the integral spum of connected graphs.

Research of Graph Compression in Information Storage

In the information storage, the image is often encountered, but the image storage and analysis are very complex, in order to more save memory space and more be used as a compressed representation of a graph, we give a new definition in the paper. We show some properties and give the lower integral sum number of some graph. DOI: http://dx.doi.org/10.11591/telkomnika.v11i8.3036

A New Graph Compression in Information Storage

In the information storage, the image is often encountered, but the image storage and analysis are very complex, in order to more save memory space and more be used as a compressed representation of a graph, we give a new definition in the paper. We show some properties and give the lower integral sum number of some graph.

Integral Sum Number of Complete Hypergraphs and Hypercycles

Integral Sum Number of Complete Hypergraphs and Hypercycles

On integral sum labeling of dense graphs

A graph is said to be a \textit{sum graph} if there exists a set $S$ of positive integers as its vertex set with two vertices adjacent whenever their sum is in $S$. An integral sum graph is defined just as the sum graph, the difference being that the label set $S$ is a subset of $Z$ instead of set of positive integers. The sum number of a given graph $G$ is defined as the smallest number of isolated vertices which when added to $G$ results in a sum graph. The integral sum number of $G$ is analogous. In this paper, we mainly prove that any connected graph $G$ of order $n$ with at least three vertices of degree $(n-1)$ is not an integral sum graph. We characterise the integral sum graph $G$ of order $n$ having exactly two vertices of degree $(n-1)$ each and hence give an alternative proof for the existence theorem of sum graphs.

The (mod, integral) sum numbers of fans and [formula omitted

Let N ( Z ) denote the set of all positive integers (integers). The sum graph G + ( S ) of a finite subset S ⊂ N ( Z ) is the graph ( S , E ) with uv ∈ E if and only if u + v ∈ S . A graph G is said to be an (integral) sum graph if it is isomorphic to the sum graph of some S ⊂ N ( Z ) . The (integral) sum number σ ( G ) ( ζ ( G ) ) of G is the smallest number of isolated vertices which when added to G result in an (integral) sum graph. A mod sum graph is a sum graph with S ⊂ Z m ⧹ { 0 } and all arithmetic performed modulo m where m ⩾ | S | + 1 . The mod sum number ρ ( G ) of G is the least number ρ of isolated vertices ρ K 1 such that G ∪ ρ K 1 is a mod sum graph. In this paper, we prove that for n ⩾ 3 , the n spoked fan F n is an integral sum graph, ρ ( F 4 ) = 1 , ρ ( F n ) = 2 for n ≠ 4 , and σ ( F n ) = 2 , n = 4 , 3 , n = 3 or n ⩾ 6 and n even , 4 , n ⩾ 5 and n odd . We also show that for K n , n - E ( nK 2 ) ( n ⩾ 6 ) , ρ = n - 2 , σ = 2 n - 3 and ζ = 2 n - 5 .

The (integral) sum number of Kn− E( Kr)

The concept of the (integral) sum graphs was introduced by Harary (Congr. Numer. 72 (1990) 101; Discrete Math. 124 (1994) 99). Let N( Z) denote the set of all positive integers(integers). The (integral) sum graph of a finite subset S⊂ N( Z) is the graph ( S, E) with two vertices that are adjacent whenever their sum is in S. A graph G is said to be a (integral) sum graph if it is isomorphic to the (integral) sum graph of some S⊂ Z. The (integral) sum number of a given graph G, denoted by σ( G)( ζ( G)), was defined as the smallest number of isolated vertices which when added to G resulted in a (integral) sum graph. In this paper, the integral sum number and the graph ( K n − E( K r )), which is an unsolved problem proposed by Harary (1994) in 1994, are investigated and determined. The results on the integral sum number of graph ( K n − E( K r )) are presented as follows: ζ(K n−E(K r))= 0 (r=n,n−1), n−2 (r=n−2), n−1 (n−3⩾r⩾⌈ 2n 3 ⌉−1), 3n−2r−4 (⌈ 2n 3 ⌉−1>r⩾ n 2 ), 2n−4 (⌈ 2n 3 ⌉−1⩾ n 2 >r⩾2), where n⩾5, r⩾2. Furthermore, if r≠ n−1, then σ( K n − E( K r ))= ζ( K n − E( K r )).

Some results on integral sum graphs

Let Z denote the set of all integers. The integral sum graph of a finite subset S of Z is the graph (S,E) with vertex set S and edge set E such that for u,v∈S, uv∈E if and only if u+v∈S. A graph G is called an integral sum graph if it is isomorphic to the integral sum graph of some finite subset S of Z. The integral sum number of a given graph G, denoted by ζ(G), is the smallest number of isolated vertices which when added to G result in an integral sum graph. Let ⌈x⌉ denote the least integer not less than the real x. In this paper, we (i) determine the value of ζ(K n−E(K r)) for r⩾⌈2n/3⌉−1, (ii) obtain a lower bound for ζ(K n−E(K r)) when 2⩽r<⌈2n/3⌉−1 and n⩾5, showing by construction that the bound is sharp when r=2, and (iii) determine the value of ζ(K r,r) for r⩾2. These results provide partial solutions to two problems posed by Harary (Discrete Math. 124 (1994) 101–108). Finally, we furnish a counterexample to a result on the sum number of K r,s given by Hartsfiedl and Smyth (Graphs and Matrices, R. Rees (Ed.), Marcel, Dekker, New York, 1992, pp. 205–211).

The integral sum number of complete bipartite graphs Kr, s

A graph G=( V, E) is said to be an integral sum graph (sum graph) if its vertices can be given a labeling with distinct integers (positive integers), so that uv∈ E if and only if u+ v∈ V. The integral sum number (sum number) of a given graph G, denoted by ζ( G) ( σ( G)), was defined as the smallest number of isolated vertices which when added to G result in an integral sum graph (sum graph). In this paper, we shall introduce a new definition of the proper r-partition of the positive integer s on a positive integer r ( s⩾ r). A partition ( s 1, s 2,…, s k ) of the positive integer s(⩾ r⩾1) is said to be a proper r-partition if it satisfies the following three conditions: (1) s= s 1+ s 2+⋯+ s k ; (2) s 1⩾1, s i⩾s i−1+r−1 (i=2,3,…,k) ; (3) s k is minimum satisfying conditions (1) and (2). Using the definition, the integral sum number and the sum number of the complete bipartite graphs K r, s , which is an unsolved problem proposed by Harary are investigated and determined. The results on the integral sum number and sum number of graphs K r, s ( s⩾ r⩾2) are presented as follows: σ(K r,s)=ζ(K r,s)=s k+r−1, where s k is the last term of the proper r-partition of the integer s. Besides, in this paper, we also obtain an analytical method which is able to find s k for any positive integers s⩾ r and we point out that the result σ( K r, s )=⌈(3 r+ s−3)/2⌉, obtained by Hartsfield and Smyth (Graphs and Matrices, Marcel Dekker, New York, 1992, pp. 205–211), is not true.

The sum number and integral sum number of complete bipartite graphs

Let N denote the set of positive integers. The sum graph G +(S) of a finite subset S of N is the graph (S,E) with vertex set S and edge set E such that for u,v∈S, uv∈E if and only if u+v∈S. A graph G is called a sum graph if it is isomorphic to the sum graph G +(S) of some finite subset S of N. The sum number σ(G) of a graph G is defined as the smallest nonnegative integer m for which G∪mK 1 is a sum graph. Let Z be the set of all integers. By extending the set N to Z in the above definitions of sum graphs and sum numbers, Harary [3] introduced the corresponding notions of integral sum graphs and integral sum number of a graph. In this paper, we evaluate the value of the sum number and integral sum number of the complete bipartite graph K r,s . While the former one corrects the result given in [4], the latter settles completely a problem proposed in [3].

On the sum number and integral sum number of hypertrees and complete hypergraphs

A hypergraph H is a sum hypergraph iff there are a finite S⊂ N + and d 1,d 2∈ N + with 1< d 1⩽ d 2<2 d 1 such that H is isomorphic to the hypergraph H d 1,d 2 +(S)=(V, E) where V≔ S and E≔ {x 1,x 2,…,x k} : k∈{d 1,d 1+1,…,d 2}∧(i≠j⇒x i≠x j)∧ ∑ i=1 k x i∈S . For an arbitrary hypergraph H , the sum number σ=σ( H) is defined to be the minimum number of isolated vertices w 1,…, w σ ∉ V such that H∪{w 1,…,w σ} is a sum hypergraph. For S⊂ Z we obtain the concept of integral sum hypergraphs and the integral sum number ζ( H). In this paper, we determine σ( K n d) and give bounds for ζ( K n d) , where K n d denotes the d-uniform complete hypergraph on n vertices. Moreover, for d⩾3 we prove that every d-uniform hypertree H is an integral sum hypergraph, i.e. ζ( H)=0 .

On integral sum graphs

A graph is said to be a sum graph if there exists a set S of positive integers as its node set, with two nodes adjacent whenever their sum is in S. An integral sum graph is defined just as the sum graph, the difference being that S is a subset of Z instead of N ∗. The sum number of a given graph G is defined as the smallest number of isolated nodes which when added to G result in a sum graph. The integral sum number of G is analogous. We are interested in some problems and conjectures posed by Harary (1994) in [4], mainly proving the conjecture that the integral sum numbers equal sum numbers for all complete graphs with at least four nodes and disproving the conjecture that every tree T with integral sum number ζ ( T) = 0 is a caterpillar. In addition, we also generalize some results of [4].

Harary's conjectures on integral sum graphs

Let N denote the set of positive integers and Z denote all integers. The (integral) sum graph of a finite subset S ⊂ N( Z) is the graph ( S, E) with uv ϵ E if and only if u + v ϵ S. A graph G is said to be an (integral) sum graph if it is isomorphic to the (integral) sum graph of some S ⊂ N( Z). The (integral) sum number of a given graph G is the smallest number of isolated nodes which when added to G result in an (integral) sum graph. We show that the integral sum number of a complete graph with n ⩾ 4 nodes equals 2 n − 3, which proves a conjecture of Harary. And we disprove another conjecture of Harary by showing that there are infinitely many trees which are not caterpillars but are integral sum graphs.

Sum graphs over all the integers

We introduced the sum graph of a set S of positive integers as the graph G +( S) having S as its node set, with two nodes adjacent whenever their sum is in S. Now we study sum graphs over all the integers so that S may contain positive or negative integers on zero. A graph so obtained is called an integral sum graph. The sum number of a given graph G was defined as the smallest number of isolated nodes which when added to G result in a sum graph. The integral sum number of G is analogous. We see that all paths and all matchings are integral sum graphs. We find the integral sum number of the small graphs and offer several intriguing unsolved problems.