book

Recent Advancements in Graph Theory

by N. P. Shrimali, Nita H. Shah

November 2020

Intermediate to advanced

410 pages

8h 8m

English

CRC Press

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1.1 Introduction1.2 Main Results1.3 Concluding Remark

2.1 Essential Preface2.2 α-absolute Mean Graceful Graphs Consisting of the Property of Universality2.2.1 Extreme Vertices in α-absolute Mean Graceful Graph2.2.2 Universality in the Path Graph2.2.3 Universality in the Star Graph2.2.4 Universality in the Coconut Tree Graph2.2.5 Special Graph Consisting of Universality in Absolute Mean Graceful Labeling
3.1 Introduction3.2 Universal α-graceful Gear related Graphs
4.1 Introduction4.2 Definitions4.3 Labeling Number of Generalized Jahangir Graph4.4 Conclusion
5.1 Introduction5.2 Basic Definitions5.3 Existing Results5.4 Main Results5.5 Concluding Remarks
6.1 Introduction6.2 Main Results6.3 Conclusions
7.1 Introduction7.2 Edge Product Cordial Labeling7.3 Total Edge Product Cordial Labeling7.4 Results related to Shadow Graph of Wheel7.5 Results related to Splitting Graph of Wheel7.6 Results related to Middle Graph of Wheel7.7 Results related to Total Graph of Wheel7.8 Concluding Remarks7.9 Open Problems
8.1 Introduction8.2 Main Results8.3 Concluding Remark
9.1 Introduction9.2 Sum Divisor Cordial Labeling for Vertex Switching of Cycle related Graphs9.3 Concluding Remarks
10.1 Introduction10.1.1 Definition10.2 Main Results10.3 Conclusion
11.1 Introduction11.2 Notations and Definitions11.3 Main Results11.4 Conclusion
12.1 Introduction and Definitions12.2 Main Results
13.1 Introduction13.1.1 Spiral Ordering of the Gaussian Integers13.1.2 Origami Models13.2 Main Results13.3 Concluding Remark
14.1 Introduction14.2 Calculation of Magic Constant14.3 The Product CN ⊗ Cn Possesses n Different Cycles of Length n14.4 Known Results and Methods
15.1 Introduction15.2 Magic Labeling15.3 Antimagic Labeling15.4 Some Known Results of Antimagic Labeling15.5 Our Investigations15.6 Conclusion and Scope for Further Research
16.1 Introduction16.2 Cartesian Product of C3t and C416.3 Direct Product of C3t and C416.4 Strong Product of C3t and C416.5 Corona Product of C4 and C3t16.6 Concluding Remark
17.1 Unary Operations on Graphs17.2 Binary Operations on Subgraphs17.3 Graphs from Subgraphs17.3.1 Graphs from Type 1 Adjacency17.3.2 Graphs from Type 2 Adjacency17.3.3 Graphs from Type 4 Adjacency17.3.4 Graphs from Type 6 Adjacency17.3.5 Subgraphs of the Line Graph17.4 Variations in the Adjacency Definition17.4.1 Union Adjacency17.4.2 Chord Crossing Adjacency17.4.3 Incidence Adjacency17.4.4 Counting based Adjacency17.4.5 Distance based Adjacency17.5 Transformation Graphs17.6 Iteration, Convergence and Completion17.7 Research Problems
18.1 Unit Graph18.1.1 Rings R for which γ(G(R))=|R|218.1.2 Rings R for which γ(G(R))=|R|−1218.2 Domination Number of Line Graph of Unit Graph …18.2.1 Line Graph of Unit Graph
19.1 Pendant Number of Cartesian Product of Graphs19.2 Pendant Number of Direct Product of Graphs19.3 Conclusion
20.1 Introduction20.2 Wiener Index of Tensor Product of Cycle Graph and Bipartite Graph20.3 Wiener Index of Tensor Product of Cycle Graph with Some Non-bipartite Graphs
21.1 Introduction21.2 Wiener Index of the Zero-divisor Graph G(Zn)21.3 Wiener Index of Zero-Divisor Graphs from Direct Product of Rings21.4 Concluding Remark
22.1 Algebraic Graph22.1.1 Existing Algebraic Graphs22.2 Signed Graph and its Parameters22.2.1 Criteria for Balance22.2.2 Criteria for Consistency22.2.3 Criteria for Sign-compatibility22.3 Algebraic Signed Graph22.3.1 Existing Algebraic Signed Graphs22.4 Signed Total Graphs22.4.1 Balanced Signed Total Graphs22.5 Signed Unit Graphs
23.1 Introduction and Preliminaries23.2 Nullity of Complete Tripartite Graphs23.2.1 Techniques for Nullity of Graph23.2.1.1 Zero-sum Weighting Technique23.2.1.2 Co-neighbor Technique23.3 Energy of Complete Tripartite Graphs23.4 Conclusion
24.1 Introduction24.2 Preliminaries24.3 Characterization of Restrained Edge Dominating Set24.4 Restrained Edge Domination Number of Star related Graphs24.5 Restrained Edge Domination Number of Cycle related Graphs24.6 Concluding Remarks
25.1 Introduction25.2 Rainbow Neighbourhoods in Graphs25.2.1 Rainbow Neighbourhood Number of Some Basic Graph Classes25.2.2 Rainbow Neighbourhood Number of Graph Operations25.2.3 Important Observations25.2.4 Rainbow Neighbourhood Number of Some Cycle related Graphs25.2.5 Rainbow Neighbourhood Number of Some Graph Transformations25.2.6 Rainbow Neighbourhood Number of Graph Products25.3 J -Coloring of Graphs25.3.1 J -colorability of Graph Operations25.3.2 The Paucity Number of Graphs25.4 Conclusion
26.1 Introduction26.2 Bounds for Total Global Dominator Chromatic Number26.3 Total Global Dominator Chromatic Number for Some Classes of Graphs26.4 Conclusion
27.1 Introduction27.2 Rainbow Vertex Connection Number in Selected Graphs27.3 Conclusions
28.1 Introduction28.2 Preliminaries28.3 A Lower Bound for hc(T )28.4 Theorem 1 vs. [3, Theorem 4]28.4.1 Special Trees A2k and A2k+128.4.2 Broom Trees B2k and B2k+128.5 Concluding Remarks
29.1 Introduction29.2 Main Results29.3 Conclusion
30.1 Introduction30.2 Randić Energy of Extended m-Shadow Graph30.3 Randić Energy of Extended m-Splitting Graph30.4 Randić Energy of Double Cover of Graph30.5 Randić Energy of Extended Double Cover of Graph30.6 Concluding Remarks
31.1 Introduction31.2 L-spectra of Graph Obtained by Duplicating Each Vertex by an Edge31.3 L-spectra of Graph Obtained by Duplicating Each Edge by a Vertex31.4 Conclusion

Content preview from Recent Advancements in Graph Theory

20Wiener Index of Tensor Product of Cycle Graph and Some Other Graphs

H. S. Mehta

Department of Mathematics,Sardar Patel University,Vallabh Vidhyanagar,Anand, Gujarat (INDIA).E-mail: hs_mehta@spuvvn.edu

J. George

Shri Alpesh N. Patel P. G. Institute of Science & Research,Anand, Gujarat (INDIA).E-mail: johnsongeorge157@gmail.com

The tensor product of graphs is a very well-known graph product and has been studied in detail. Also the Wiener index of a connected graph G is defined as, $W (G) = \sum_{{x, x^{'}} \subset V (G)} d (x, x^{'})$ . In this chapter, we obtain W(C_2m+1 ⊗ H) for any connected, bipartite graph H with diameter at most 4. Also, we obtain W(C_2m+1 ⊗ H) for some non-bipartite graphs H as the wheel graph W_n, Helm graph H_n etc.