Distribution Number of the Kronecker and Vertex Gluing of Graphs
Distribution Number of the Kronecker
Product and Vertex Gluing of Graphs
jvb
A RESEARCH PROPOSAL
Presented to
The Research Director of
Siquijor State College
Siquijor, Siquijor, Philippines
jvb
Proponents
Dr. Ruben A. Sanchez 1
Dr. Baldomero R. Martinez 2
Dr. Michael P. Baldado Jr. 3
October 2012
1
Siquijor State College, Philippines
Siquijor State College, Philippines
3
Research Consultant, Negros Oriental State University, Philippines
2
TABLE OF CONTENTS
TITLE PAGE
i
TABLE OF CONTENTS
ii
LIST OF NOTATIONS
iii
LIST OF FIGURES
iv
1 INTRODUCTION
1.1 Objectives of the Study . . . . . . . . . .
1.2 Significance of the Study . . . . . . . . .
1.3 Scope and Limitations . . . . . . . . . .
1.4 Review of Related Literature and Studies
2 BASIC CONCEPTS
REFERENCES
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7
21
LIST OF NOTATIONS
Symbol
|S|
G
Kn
Km,n
G◦H
Cn σ(G) rad(G) diam(G) d(u, v)
E(G)
uv
Fn
Wn
G+H
NG (x)
V (G)
|V (G)|
Pn
S
K1
˙
X ∪Y
C3 = [1, 2, 3]
Name cardinality of set S complement of G complete graph of order n complete bipartite graph of order m + n corona of G and H cycle of length n
CDPU number of G radius of G diameter of G distance of vertices u and v edge set of G edge joining vertices u and v fan of order n + 1 wheel of order n + 1 join of G and H neighborhood of x in G vertex set of G order of G path of length n − 1 subgraph induced by S trivial graph G disjoint union cycle of order 3
Page
27
17
14
15
20
14
37
37
37
37
11
12
17
18
17
16
11
11
13
13
14
13
14
LIST OF FIGURES
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
2.20
A graph G . . . . . . . . . . . . . . . . . . . . . .
A graph H which is an
Product and Vertex Gluing of Graphs
jvb
A RESEARCH PROPOSAL
Presented to
The Research Director of
Siquijor State College
Siquijor, Siquijor, Philippines
jvb
Proponents
Dr. Ruben A. Sanchez 1
Dr. Baldomero R. Martinez 2
Dr. Michael P. Baldado Jr. 3
October 2012
1
Siquijor State College, Philippines
Siquijor State College, Philippines
3
Research Consultant, Negros Oriental State University, Philippines
2
TABLE OF CONTENTS
TITLE PAGE
i
TABLE OF CONTENTS
ii
LIST OF NOTATIONS
iii
LIST OF FIGURES
iv
1 INTRODUCTION
1.1 Objectives of the Study . . . . . . . . . .
1.2 Significance of the Study . . . . . . . . .
1.3 Scope and Limitations . . . . . . . . . .
1.4 Review of Related Literature and Studies
2 BASIC CONCEPTS
REFERENCES
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.
.
.
1
2
2
2
3
7
21
LIST OF NOTATIONS
Symbol
|S|
G
Kn
Km,n
G◦H
Cn σ(G) rad(G) diam(G) d(u, v)
E(G)
uv
Fn
Wn
G+H
NG (x)
V (G)
|V (G)|
Pn
S
K1
˙
X ∪Y
C3 = [1, 2, 3]
Name cardinality of set S complement of G complete graph of order n complete bipartite graph of order m + n corona of G and H cycle of length n
CDPU number of G radius of G diameter of G distance of vertices u and v edge set of G edge joining vertices u and v fan of order n + 1 wheel of order n + 1 join of G and H neighborhood of x in G vertex set of G order of G path of length n − 1 subgraph induced by S trivial graph G disjoint union cycle of order 3
Page
27
17
14
15
20
14
37
37
37
37
11
12
17
18
17
16
11
11
13
13
14
13
14
LIST OF FIGURES
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
2.20
A graph G . . . . . . . . . . . . . . . . . . . . . .
A graph H which is an
References: Buczkowski, P.S., Chartrand, G., Poisson, C. & Zhang,P.(2003). (2005). On the metric dimension of some families of graphs, Electronic Notes in Discrete Math., 22 , 129-133.134 Chappell,G.G., Gimbel,J. & Hartman,C.(2010). Bounds on the metric Chartrand G., Eroh,L., Johnson,M.A., & Oellermann,O.R.(2000) Chartrand G., & Lesniak,L.(2000). Graphs and Digraphs, 3rd ed., Chapman and Hall/CRC,. Fehr M., Gosselin,S. & Oellermann,O.R.(2006). The metric dimension of Cayley digraphs, Discrete Math., 306, 31-41. Harary F. & Melter,R.A.(1976). On the metric dimension of a graph, Ars. Iswadi H., Baskoro,E.T., Simanjuntak,R., & Salman, A.N.M.(2012). The metric dimension of graph with pendant edges, URL:repository. Slater, P.J.(1975)., Leaves of trees, Congr. Numer., 14 , 549-559. 22 Slater, P.J.(1988)., Dominating and reference sets in graphs, J