Comparing the irregularity and the total irregularity of graphs (original) (raw)
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Abstract
has defined the irregularity of a simple undirected graph G as irr , where d G (u) denotes the degree of a vertex u ∈ V (G). Recently, in a new measure of irregularity of a graph, so-called the total irregularity, was defined as irr t Here, we compare the irregularity and the total irregularity of graphs. For a connected graph G with n vertices, we show that irr t (G) ≤ n 2 irr(G)/4. Moreover, if G is a tree, then irr t (G) ≤ (n -2)irr(G).
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References (19)
- H. Abdo, S. Brandt and D. Dimitrov, The total irregularity of a graph, Discrete Math. Theor. Comput. Sci. 16 (2014), 201-206.
- Y. Alavi, A. Boals, G. Chartrand, P. Erdős and O. R. Oellermann, k-path irregular graphs, Congr. Numer. 65 (1988) 201-210.
- Y. Alavi, G. Chartrand, F. R. K. Chung, P. Erdős, R. L. Graham and O. R. Oellermann, Highly irregular graphs, J. Graph Theory 11 (1987) 235-249.
- M. O. Albertson, The irregularity of a graph, Ars Comb. 46 (1997) 219-225.
- F. K. Bell, A note on the irregularity of graphs, Linear Algebra Appl. 161 (1992) 45-54.
- Y. Caro and R. Yuster, Graphs with large variance, Ars Comb. 57 (2000) 151-162.
- G. Chartrand, P. Erdős and O. R. Oellermann, How to define an irregular graph, Coll. Math. J. 19 (1988) 36-42.
- G. Chartrand, K. S. Holbert, O. R. Oellermann and H. C. Swart, F -degrees in graphs, Ars Comb. 24 (1987) 133-148.
- L. Collatz and U. Sinogowitz, Spektren endlicher Graphen, Abh. Math. Sem. Univ. Hamburg 21 (1957) 63-77.
- D. Cvetković and P. Rowlinson, On connected graphs with maximal index, Publications de l'Institut Mathematique (Beograd) 44 (1988) 29-34.
- D. Dimitrov, T. Réti, Graphs with equal irregularity indices, Acta Polytech. Hung. 11 (2014), 41-57.
- P. Erdős and T. Gallai, Graphs with prescribed degrees of vertices, (in Hungarian) Mat. Lapok. 11 (1960) 264-274.
- P. Hansen and H. Mélot, Variable neighborhood search for extremal graphs 9. Bounding the irregularity of a graph, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 69 (1962) 253-264.
- M. A. Henning and D. Rautenbach, On the irregularity of bipartite graphs, Discrete Math. 307 (2007) 1467-1472.
- D. E. Jackson and R. Entringer, Totally segregated graphs, Congress. Numer. 55 (1986) 159- 165.
- W. Luo and B. Zhou, On the irregularity of trees and unicyclic graphs with given matching number, Util. Math. 83 (2010) 141-147.
- D. Rautenbach and I. Schiermeyer, Extremal problems for imbalanced edges, Graphs Comb. 22 (2006) 103-111.
- D. Rautenbach and L. Volkmann, How local irregularity gets global in a graph, J. Graph Theory 41 (2002) 18-23.
- B. Zhou and W. Luo, On irregularity of graphs, Ars Combin. 88 (2008) 55-64.
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