On the difference between the revised Szeged index and the Wiener index

Let Sz^@?(G) and W(G) be the revised Szeged index and the Wiener index of a graph G. Chen et al. (2014) proved that if G is a non-bipartite connected graph of order n>=4, then Sz^@?(G)-W(G)>=(n^2+4n-6)/4. Using a matrix method we prove that if G is a non-bipartite graph of order n, size m, and girth g, then Sz^@?(G)-W(G)>=n(m-3n4)+P(g), where P is a fixed cubic polynomial. Graphs that attain the equality are also described. If in addition g>=5, then Sz^@?(G)-W(G)>=n(m-3n4)+(n-g)(g-3)+P(g). These results extend the bound of Chen, Li, and Liu as soon as m>=n+1 or g>=5. The remaining cases are treated separately.

[1]  H. Wiener.  Structural determination of paraffin boiling points. , 1947, Journal of the American Chemical Society.

[2]  Padmakar V. Khadikar,et al.  A Novel PI Index and Its Applications to QSPR/QSAR Studies , 2001, J. Chem. Inf. Comput. Sci..

[3]  Sandi Klavzar and Mohammad J. Nadjafi-Arani.  Cut Method: Update on Recent Developments and Equivalence of Independent Approaches , 2014 .

[4]  Sandi Klavzar,et al.  On the canonical metric representation, average distance, and partial Hamming graphs , 2006, Eur. J. Comb..

[5]  Bo Zhou,et al.  On the revised Szeged index , 2011, Discret. Appl. Math..

[6]  Aleksandra Tepeh,et al.  Orientations of graphs with maximum Wiener index , 2016, Discret. Appl. Math..

[7]  Bojan Mohar,et al.  Labeling of Benzenoid Systems which Reflects the Vertex-Distance Relations , 1995, J. Chem. Inf. Comput. Sci..

[8]  Sandi Klavzar,et al.  An Algorithm for the Calculation of the Szeged Index of Benzenoid Hydrocarbons , 1995, J. Chem. Inf. Comput. Sci..

[9]  Hanyuan Deng,et al.  The (n,n)-graphs with the first three extremal Wiener indices , 2008 .

[10]  Tomaz Pisanski,et al.  Edge-contributions of some topological indices and arboreality of molecular graphs , 2009, Ars Math. Contemp..

[11]  I. Gutman,et al.  Wiener Index of Trees: Theory and Applications , 2001 .

[12]  Xueliang Li,et al.  Bicyclic graphs with maximal revised Szeged index , 2013, Discret. Appl. Math..

[13]  Xueliang Li,et al.  The (revised) Szeged index and the Wiener index of a nonbipartite graph , 2014, Eur. J. Comb..

[14]  Martin Knor,et al.  Wiener index of iterated line graphs of trees homeomorphic to , 2013, Discret. Math..

[15]  Sandi Klavzar,et al.  Improved bounds on the difference between the Szeged index and the Wiener index of graphs , 2014, Eur. J. Comb..

[16]  Mohammad Ali Hosseinzadeh,et al.  The Relationships between Wiener Index, Stability Number and Clique Number of Composite Graphs , 2013 .

[17]  M. J. Nadjafi-Arani,et al.  Extremal graphs with respect to the vertex PI index , 2009, Appl. Math. Lett..

[18]  Frank Harary,et al.  Distance in graphs , 1990 .

[19]  Mustapha Aouchiche,et al.  On a conjecture about the Szeged index , 2010, Eur. J. Comb..

[20]  Milan Randić.  On generalization of wiener index for cyclic structures , 2002 .

[21]  Douglas F. Rall,et al.  Canonical Metric Representation , 2008 .

[22]  Ali Reza Ashrafi,et al.  On the differences between Szeged and Wiener indices of graphs , 2011, Discret. Math..

[23]  W. Imrich,et al.  Handbook of Product Graphs, Second Edition , 2011 .

[24]  Martin Knor,et al.  Wiener index of iterated line graphs of trees homeomorphic to the claw K1, 3 , 2013, Ars Math. Contemp..

[25]  Kinkar Ch. Das,et al.  Bound for vertex PI index in terms of simple graph parameters , 2013 .

[26]  Shuchao Li,et al.  On the sum of all distances in bipartite graphs , 2014, Discret. Appl. Math..

[27]  D. Djoković.  Distance-preserving subgraphs of hypercubes , 1973 .

[28]  Mustapha Aouchiche,et al.  The Normalized Revised Szeged Index , 2011 .

[29]  I. Gutman,et al.  Mathematical Concepts in Organic Chemistry , 1986 .

[30]  Tomaz Pisanski,et al.  Use of the Szeged index and the revised Szeged index for measuring network bipartivity , 2010, Discret. Appl. Math..

[31]  Ali Reza Ashrafi,et al.  Graphs whose Szeged and Wiener numbers differ by 4 and 5 , 2012, Math. Comput. Model..

[32]  Roberto Cruz,et al.  Wiener index of Eulerian graphs , 2014, Discret. Appl. Math..

[33]  Sandi Klavžar,et al.  The Szeged and the Wiener Index of Graphs , 1996 .

[34]  I. Gutman,et al.  Wiener Index of Hexagonal Systems , 2002 .

[35]  Sandi Klavzar,et al.  Wiener index versus Szeged index in networks , 2013, Discret. Appl. Math..

[36]  Ali Reza Ashrafi,et al.  Revised and edge revised Szeged indices of graphs , 2014, Ars Math. Contemp..

[37]  Aleksandar Ilic,et al.  Note on PI and Szeged indices , 2010, Math. Comput. Model..

[38]  Ali Reza Ashrafi,et al.  On Extensions of Wiener Index , 2013 .

[39]  Ali Reza Ashrafi,et al.  Further results on hierarchical product of graphs , 2013, Discret. Appl. Math..

[40]  A. R. ASHRAFI,et al.  A NEW PROOF OF THE SZEGED – WIENER THEOREM , 2011 .

[41]  Dragan Stevanovic,et al.  Bicyclic graphs with extremal values of PI index , 2013, Discret. Appl. Math..

[42]  Xueliang Li,et al.  On a relation between Szeged and Wiener indices of bipartite graphs , 2012 .

[43]  Shuchao Li,et al.  On the further relation between the (revised) Szeged index and the Wiener index of graphs , 2016, Discret. Appl. Math..