Some results on harmonic mean graphs

Abstract

A graph G=(V,E) with p vertices and q edges is called a harmonic mean graph if it is possible to label the vertices x∈V with distinct labels f(x) from 1,2,...,q+1 in such a way that when each edge e=uv is labeled with f(uv)=2f(u)f(v) f(u)+f(v) or 2f(u)f(v) f(u)+f(v) then the edge labels are distinct. In this case f is called harmonic mean labeling of G. The concept of Harmonic mean labeling was introduced in [the authors, “Harmonic mean labelings of graphs” (to appear)]. In [loc. cit.] and in [Int. J. Contemp. Math. Sci. 7, No. 1–4, 197–208 (2012; Zbl 1251.05128)] we investigate the harmonic mean labeling of several standard graphs such as path, cycle comb, ladder, triangular snakes, quadrilateral snakes etc. In the present paper, we investigate the harmonic mean labeling for a polygonal chain, square of the path and dragon. Also we enumerate all harmonic mean graph of order ≤5.

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www.ccsenet.org/jmr Journal of Mathematics Research Vol. 4, No. 1; February 2012
Some More Results on Harmonic Mean Graphs
S. S. Sandhya (Corresponding author)
Department of Mathematics, Sree Ayyappa College for Women
Chunkankadai, Kanyakumari 629 807, Tamilnadu, India
E-mail: sssandhya2009@gmail.com
S. Somasundaram
Department of Mathematics, Manonmaniam Sundaranar University
Tirunelveli 627 012, Tamilnadu, India
E-mail: somumsu@rediffmail.com
R. Ponraj
Department of Mathematics, Sri Paramakalyani College
Alwarkurichi 627 412, Tamilnadu, India
E-mail: ponrajmath@gmail.com
Received: November 10, 2011 Accepted: November 28, 2011 Published: February 1, 2012
doi:10.5539/jmr.v4n1p21 URL: http://dx.doi.org/10.5539/jmr.v4n1p21
Abstract
A Graph G=(V,E) with pvertices and qedges is called a harmonic mean graph if it is possible to label the vertices x∈V
with distinct labels f(x) from 1,2, . . . , q+1 in such a way that when each edge e=uv is labeled with f(uv)=⌈2f(u)f(v)
f(u)+f(v)⌉
or ⌊2f(u)f(v)
f(u)+f(v)⌋then the edge labels are distinct. In this case fis called Harmonic mean labeling of G.
The concept of Harmonic mean labeling was introduced in (Somasundaram, Ponraj & Sandhya). In (Somasundaram,
Ponraj & Sandhya) and (Sandhya, Somasundaram & Ponraj, 2012) we investigate the harmonic mean labeling of several
standard graphs such as path, cycle comb, ladder, Triangular snakes, Quadrilateral snakes etc. In the present paper, we
investigate the harmonic mean labeling for a polygonal chain, square of the path and dragon. Also we enumerate all
harmonic mean graph of order ≤5.
Keywords: Graph, Harmonic mean graph, Polygonal chain, Square of a path, Dragon
AMS subject classification: 05C 78
1. Introduction
The graph considered here will be finite, undirected and simple. Terms not defined here are used in the sense of Harary
(Harary, 1988). The symbols V(G) and E(G) will denote the vertex set and edge set of a graph G. The Cardinality of
the graph Gis called the order of G. The Cardinality of its edge set is called the size of G. The graph G−eis obtained
from Gby deleting an edge e. The sum G1+G2of two graphs G1and G2has vertex set V(G1)∪V(G2) and edge set
E(G1+G2)=E(G1)∪E(G2)∪ {uv :u∈V(G1) and v∈V(G2)}. The Union of two graphs G1and G2is a graph G1∪G2
with vertex set V(G1∪G2)=V(G1)∪V(G2) and E(G1∪G2)=E(G1)∪E(G2). The Square G2of the graph Ghas V(G2)
with u,vadjacent in G2whenever d(u,v)≤2 in the graph G. A dragon is a graph formed by joining an end vertex of path
Pmto the vertex of the cycle Cn. It is denoted Cn@Pm. The graph Cnˆ◦K1,mis obtained Cnand K1,mby identifying any
vertex of Cnand the central vertex of K1,m.
S. Somasundaram and R. Ponraj introduced mean labeling of graphs in (Somasundaram & Ponraj, 2003a) and investigate
mean labeling for some standard graphs in (Somasundaram & Ponraj, 2003b) and in (Somasundaram & Ponraj, 2004).
We introduce Harmonic mean labeling of graphs in (Somasundaram, Ponraj & Sandhya) and proved Harmonic mean
labeling of some standard graphs in (Sandhya, Somasundaram & Ponraj, 2012). In this paper we prove that polygonal
chains, square of a path and dragons are harmonic mean graphs. Finally we investigate all harmonic mean graph of order
≤5.
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We shall make frequent reference to the following results.
Theorem 1.1 (Somasundaram, Ponraj & Sandhya) Any path is a harmonic mean graph.
Theorem 1.2 (Somasundaram, Ponraj & Sandhya) Any cycle Cn, n ≥3is a harmonic mean graph.
Theorem 1.3 (Somasundaram, Ponraj & Sandhya) The complete graph the Knis a harmonic graph if and only if n ≤3.
Theorem 1.4 (Somasundaram, Ponraj & Sandhya) The complete bipartite graph K1,nis a harmonic mean graph if and
only if n ≤7.
Theorem 1.5 (Sandhya, Somasundaram & Ponraj, 2012) Any wheel Wnis not a harmonic mean graph.
Theorem 1.6 (Sandhya, Somasundaram & Ponraj, 2012) Cm∪Pn, n >1,Cm∪Cn,m,n≥3is a harmonic mean graph.
Theorem 1.7 (Sandhya, Somasundaram & Ponraj, 2012) Kn−e, n >4is not a harmonic mean graph.
Remark 1.8 (Sandhya, Somasundaram & Ponraj, 2012) If p>q+1, then the graph G=(p,q) is not a harmonic mean
graph.
Remark 1.9 (Sandhya, Somasundaram & Ponraj, 2012) If n≤4, Kn−eis a harmonic mean graph.
2. Harmonic Mean Labeling
Definition 2.1 A Graph Gwith pvertices and qedges is called a harmonic mean graph if it is possible to label the
vertices x∈Vwith distinct labels f(x) from 1,2, . . . , q+1 in such a way that when each edge e=uv is labeled with
f(uv)=⌈2f(u)f(v)
f(u)+f(v)⌉or ⌊2f(u)f(v)
f(u)+f(v)⌋then the edge labels are distinct. In this case fis called Harmonic mean labeling of G.
Definition 2.2 A Polygonal chain Gm,nis a connected graph all of whose mblocks are polygons (Cn).
Theorem 2.3 Polygonal chain Gm,nare Harmonic mean graphs for all m and n.
Proof: In Gm,nchain, let u1u2u4u6. . . un−4un−2un+1un−1un−3un−5. . . u7u5u3u1be the first cycle. The second cy-
cle is connected to the first cycle at the vertex un+1. Let un+1un+2un+4. . . u2n+1u2n−1u2n−3. . . un+7un+5un+3un+1
be the second cycle. The third chain is connected to the second cycle at the vertex
u2n+1. Let the third cycle be u2n+1u2n+2u2n+4. . . u3n+1u3n−1u3n−3. . . u2n+5u2n+3u2n+1. In general
the rth cycle is connected to the (r−1)th cycle at the vertex urn+1. Let the rth cycle be
urn+1urn+2urn+4urn+6. . . u(r+1)n−4u(r+1)n−2u(r+1)n+1u(r+1)n−1u(r+1)n−3u(r+1)n−5u(r+1)n−7. . . urn+5urn+3urn+1.
Magnified figure of the rth cycle is given in Figure 1.
Assume the graph has mcycles. Define a function f:V(Gm,n)→ {1,2, . . . , q+1}by f(vi)=i, 1 ≤i≤mn −1, f(vn)=
mn +1. Then the label of the edge is given below
f(umn+1umn+2)=mn +1
f(umn+iumn+i+2)=mn +i+1
f(u(m+1)n−2u(m+1)n+1)=(m+1)n−1
f(u(m+1)n+1u(m+1)n−1)=(m+1)n
Since the graph Gm,nhas distinct edge labels, Gm,nis a harmonic mean graph.
Example 2.4 Harmonic mean labeling of G4,9chain is in Figure 2.
Now we investigate the square of a path.
Theorem 2.5 The graph P2
nis a harmonic mean graph.
Proof: Let Pnbe the path u1,u2,...,un. Clearly P2
nhas nvertices and 2n−3 edges. Define f:V(P2
n)→ {1,2, . . . , q+1}
by f(ui)=2i−1,1≤i≤n−1 and f(un)=2n−2. The label of the edge uiui+1is 2i−1,1≤i≤n−1. The label of the
edge uiui+2is 2i,1≤i≤n−2. The label of the edge un−1unis 2n−3. Hence P2
nis a harmonic mean graph.
Example 2.6 Harmonic mean labeling of P2
8is given in Figure 3.
Theorem 2.7 Dragon’s Cn@Pmare harmonic mean graphs.
Proof: Let u1u2. . . unbe the cycle Cnand v1v2. . . vmbe the path Pm. Identify un−1with v1. Define a function f:
V(Cn@Pm)→ {1,2, . . . , q+1}by f(ui)=i,1≤i≤n−3, f(un−2)=n−1, f(un−1)=n,f(un)=n−2 and f(vi+1)=
n+i,1≤i≤m−1. Clearly fis a harmonic mean labeling
Example 2.8 A harmonic mean labeling of C6@P7is given in Figure 4.
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Theorem 2.9 The graph Cnˆ◦K1,1is a harmonic mean graph.
Proof: We know that, Cnˆ◦K1,1=Cn@P2. Hence the proof follows from the Theorem 2.7.
Remark 2.10 The graph Cnˆ◦K1,2is also a harmonic mean graph.
Theorem 2.11 Let Cnbe the cycle u1u2u3...un. Let G be the graph with V(G)=V(Cn)∪ {v1,v2}and E(G)=E(Cn)∪
{u1v1,unv2}. Then G is a harmonic mean graph.
Proof: Define a function f:V(G)→ {1,2, . . . , q+1}by f(ui)=i+1,1≤i≤n,f(v1)=1 and f(v2)=n+2. Then we
get distinct edge labels from 1,2, . . . , q. Obviously fis a harmonic mean labeling for G.
Example 2.12 The harmonic mean labeling of G is given in Figure 5.
Theorem 2.13 Let G be the graph obtained from K4−e by attaching Pnwith the vertex of degree 3 in K4−e. Then G is a
harmonic mean graph.
Proof: Let v1,v2,v3and v4be the vertices of G=K4−eand e=v1v3. Let Pnbe the path u1u2u3. . . un. Identify u1with
v4. Define f:V(G)→ {1,2, . . . , q+1}by f(v1)=1, f(v2)=3, f(v3)=4f(v4)=6, f(ui+1)=6+i, 1 ≤i≤n−1.
Obviously Gis a harmonic mean graph.
Example 2.14 Harmonic mean labeling of Gis given in Figure 6.
Next we have
Theorem 2.15 Let G be the graph obtained from K4−e by attaching Pnto the vertex of the degree 2 in the graph K4−e.
Then G is a harmonic mean graph.
Proof: Let v1,v2,v3and v4be the vertices ofG=K4−eand u1u2u3. . . unbe the path Pn. Define f:V(G)→ {1,2, . . . , q+1}
by f(v1)=1, f(v2)=3, f(v3)=5f(v4)=7, f(ui+1)=5+i, 1 ≤i≤n−1. Clearly f is a harmonic mean labeling.
Example 2.16 Harmonic mean labeling of Gis given in Figure 7.
3. Harmonic Mean Graph of Order ≤5
Theorem 3.1 The following graphs of Order ≤5are harmonic mean graphs
i) P1,P2,P3,P4,P5
ii)C3,C4,C5
iii)
iv) K1,3,K1,4
v) C3∪P2
vi)
Proof: Since all paths are harmonic mean graphs, the graphs in case (i) are harmonic mean graphs by Theorem 1.1 The
graphs in case (ii) are harmonic mean graphs by Theorem 1.2 The graph in case (iii) C3ˆ◦K1,1,C3ˆ◦K1,2and C4ˆ◦K1,1are
harmonic mean graphs by Theorem 2.9 and Remark 2.10. The graphs in case (iv) are harmonic mean graphs by Theorem
1.4. The graphs in case (v) is a harmonic mean graph by Theorem 1.6. The graph in case (vi) are harmonic mean graphs
by Remark 1.9.
Theorem 3.2 The following graphs of order ≤5are not harmonic mean graphs.
i) K2c,K3c,K4c,K5c,
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ii) K4,K5,K4∪K1
iii)W4,W5
iv) K5−e
Proof: The graphs in case (i) are not harmonic mean graph by Remark 1.8 The graphs in case (ii) are not harmonic mean
graphs by Theorem 1.3. The graphs in case (iii) are not harmonic mean graph by Theorem 1.5. The graph in case (iv) are
not harmonic mean graph by Theorem 1.7.
Remark 3.3 The remaining 16 graphs of order ≤5 are given in Table 3.1 and Table 3.2. They are shown to be harmonic
mean graphs by giving a specific harmonic mean labeling.
The Table 3.3 shows the number of graphs of order ≤5 which are harmonic mean and which are not harmonic mean.
References
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29-35.
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9, 1-3.
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Somasundaram, S., & Sandhya, S. S. (2011a). Skolem harmonic mean labelings of graphs, to appear in Bulletin of pure
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Table 3.1
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Table 3.2
Table 3.3
Order of the graph Harmonic mean Not Harmonic mean
1 1 0
2 1 1
3 2 2
4 5 5
5 22 13
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