On degree resistance distance of cacti

A graph G is called a cactus if each block of G is either an edge or a cycle. Denote by C a c t ( n ; t ) the set of connected cacti possessing n vertices and t cycles. In a recent paper (Du et?al., 2015), the C a c t ( n ; t ) with minimum degree resistance distance was characterized. We now determine the elements of C a c t ( n ; t ) with second-minimum and third-minimum degree resistance distances. In addition, some mistakes in Du et?al. (2015) are pointed out.

[1]  Jia-Bao Liu,et al.  Asymptotic incidence energy of lattices , 2015 .

[2]  Changjiang Bu,et al.  Resistance distance in subdivision-vertex join and subdivision-edge join of graphs , 2014 .

[3]  Kexiang Xu,et al.  A Survey on Graphs Extremal with Respect to Distance-Based Topological Indices , 2014 .

[4]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[5]  Heping Zhang,et al.  Some rules on resistance distance with applications , 2008 .

[6]  Dong Li,et al.  Complete characterization of bicyclic graphs with minimal Kirchhoff index , 2016, Discret. Appl. Math..

[7]  Andrés M. Encinas,et al.  A formula for the Kirchhoff index , 2008 .

[8]  Peter Dankelmann,et al.  Degree distance of unicyclic and bicyclic graphs , 2011, Discret. Appl. Math..

[9]  Douglas J. Klein,et al.  Resistance distance-based graph invariants of subdivisions and triangulations of graphs , 2014, Discret. Appl. Math..

[10]  Zuhe Zhang,et al.  Some physical and chemical indices of clique-inserted lattices , 2013, 1302.5932.

[11]  Yujun Yang,et al.  Unicyclic graphs with extremal Kirchhoff index , 2008 .

[12]  Anna Torriero,et al.  New upper and lower bounds for the additive degree-Kirchhoff index , 2013 .

[13]  Kinkar Chandra Das,et al.  Ordering connected graphs by their Kirchhoff indices , 2016, Int. J. Comput. Math..

[14]  Xing Gao,et al.  Kirchhoff index in line, subdivision and total graphs of a regular graph , 2012, Discret. Appl. Math..

[15]  Jia-Bao Liu,et al.  Asymptotic Laplacian-energy-like invariant of lattices , 2014, Appl. Math. Comput..

[16]  Jing Huang,et al.  ON THE NORMALISED LAPLACIAN SPECTRUM, DEGREE-KIRCHHOFF INDEX AND SPANNING TREES OF GRAPHS , 2015, Bulletin of the Australian Mathematical Society.

[17]  José Luis Palacios.  On the Kirchhoff index of graphs with diameter 2 , 2015, Discret. Appl. Math..

[18]  Douglas J. Klein,et al.  A recursion formula for resistance distances and its applications , 2013, Discret. Appl. Math..

[19]  Jia-Bao Liu,et al.  The Laplacian polynomial of graphs derived from regular graphs and applications , 2014, Ars Comb..

[20]  José Luis Palacios.  Resistance distance in graphs and random walks , 2001 .

[21]  Bojan Mohar,et al.  The Quasi-Wiener and the Kirchhoff Indices Coincide , 1996, J. Chem. Inf. Comput. Sci..

[22]  Luzhen Ye.  On the Kirchhoff index of some toroidal lattices , 2011 .

[23]  R. Bapat,et al.  A Simple Method for Computing Resistance Distance , 2003 .

[24]  Jia-Bao Liu,et al.  A unified approach to the asymptotic topological indices of various lattices , 2015, Appl. Math. Comput..

[25]  José Luis Palacios,et al.  Bounds for the Kirchhoff index of regular graphs via the spectra of their random walks , 2010 .

[26]  Weigen Yan,et al.  Some physical and chemical indices of the Union Jack lattice , 2015 .

[27]  José Luis Palacios.  Foster's Formulas via Probability and the Kirchhoff Index , 2004 .

[28]  Xing Gao,et al.  Resistance distances and the Kirchhoff index in Cayley graphs , 2011, Discret. Appl. Math..

[29]  Jinde Cao,et al.  The resistance distances of electrical networks based on Laplacian generalized inverse , 2015, Neurocomputing.

[30]  Yujun Yang.  The Kirchhoff index of subdivisions of graphs , 2014, Discret. Appl. Math..

[31]  José Luis Palacios,et al.  Another look at the degree-Kirchhoff index , 2010 .

[32]  Ioan Tomescu,et al.  Extremal topological indices for graphs of given connectivity , 2015 .

[33]  Lihua Feng,et al.  A note on the Kirchhoff index of bicyclic graphs , 2014, Ars Comb..

[34]  Anna Torriero,et al.  Bounds for the Kirchhoff index via majorization techniques , 2013, Journal of Mathematical Chemistry.

[35]  Jinde Cao,et al.  A note on ‘some physical and chemical indices of clique-inserted lattices’ , 2014 .

[36]  Jinde Cao,et al.  Applications of Laplacian spectra for n-prism networks , 2016, Neurocomputing.

[37]  Weigen Yan,et al.  The triangular kagomé lattices revisited , 2013 .

[38]  Alexandru I. Tomescu.  Unicyclic and bicyclic graphs having minimum degree distance , 2008, Discret. Appl. Math..

[39]  Weijun Liu,et al.  Further results regarding the degree Kirchhoff index of graphs , 2014 .

[40]  Huiqing Liu,et al.  A Unified Approach to Extremal Cacti for Different Indices , 2008 .

[41]  Guihai Yu,et al.  Degree resistance distance of unicyclic graphs , 2012 .

[42]  Ioan Tomescu.  Properties of connected graphs having minimum degree distance , 2009, Discret. Math..

[43]  José Luis Palacios.  On the Kirchhoff index of regular graphs , 2009 .

[44]  Jianhua Tu,et al.  The degree resistance distance of cacti , 2015, Discret. Appl. Math..