2019-05-21T18:55:29Z http://toc.ui.ac.ir/?_action=export&rf=summon&issue=974
2014-12-01 10.22108
Transactions on Combinatorics Trans. Comb. 2251-8657 2251-8657 2014 3 4 Randic incidence energy of graphs Ran Gu Fei Huang Xueliang Li Let \$G\$ be a simple graph with vertex set \$V(G) = {v_1, v_2,ldots, v_n}\$ and edge set \$E(G) = {e_1, e_2,ldots, e_m}\$. Similar to the Randi'c matrix, here we introduce the Randi'c incidence matrix of a graph \$G\$, denoted by \$I_R(G)\$, which is defined as the \$ntimes m\$ matrix whose \$(i,j)\$-entry is \$(d_i)^{-frac{1}{2}}\$ if \$v_i\$ is incident to \$e_j\$ and \$0\$ otherwise. Naturally, the Randi'c incidence energy \$I_RE\$ of \$G\$ is the sum of the singular values of \$I_R(G)\$. We establish lower and upper bounds for the Randic incidence energy. Graphs for which these bounds are best possible are characterized. Moreover, we investigate the relation between the Randic incidence energy of a graph and that of its subgraphs. Also we give a sharp upper bound for the Randic incidence energy of a bipartite graph and determine the trees with the maximum Randic incidence energy among all \$n\$-vertex trees. As a result, some results are very different from those for incidence energy. Randi'c incidence matrix Randi'c incidence energy eigenvalues 2014 12 01 1 9 http://toc.ui.ac.ir/article_5573_68f2261c2087d1f09fb34c2f8de4b053.pdf
2014-12-01 10.22108
Transactions on Combinatorics Trans. Comb. 2251-8657 2251-8657 2014 3 4 On Lict sigraphs Veena Mathad Kishori Narayankar A signed graph (marked graph) is an ordered pair \$S=(G,sigma)\$‎ ‎\$(S=(G,mu))\$‎, ‎where \$G=(V,E)\$ is a graph called the underlying‎ ‎graph of \$S\$ and \$sigma:Erightarrow{+,-}\$‎ ‎\$(mu:Vrightarrow{+,-})\$ is a function‎. ‎For a graph \$G\$‎, ‎\$V(G)‎, ‎E(G)\$ and \$C(G)\$ denote its vertex set‎, ‎edge set and cut-vertex‎ ‎set‎, ‎respectively‎. ‎The lict graph \$L_{c}(G)\$ of a graph \$G=(V,E)\$‎ ‎is defined as the graph having vertex set  \$E(G)cup C(G)\$ in which‎ ‎two vertices are adjacent if and only if they correspond to‎ ‎adjacent edges of \$G\$ or one corresponds to an edge \$e_{i}\$ of \$G\$‎ ‎and the other corresponds to a cut-vertex \$c_{j}\$ of \$G\$ such that‎ ‎\$e_{i}\$ is incident with \$c_{j}\$‎. ‎In this paper‎, ‎we introduce lict‎ ‎sigraphs‎, ‎as a natural extension of the notion of lict graph to‎ ‎the realm of signed graphs‎. ‎We show that every lict sigraph is‎ ‎balanced‎. ‎We characterize signed graphs \$S\$ and \$S^{'}\$ for which‎ ‎\$Ssim L_{c}(S)\$‎, ‎\$eta(S)sim L_{c}(S)\$‎, ‎\$L(S)sim L_{c}(S')\$‎, ‎\$J(S)sim L_{c}(S^{'})\$ and \$T_{1}(S)sim L_{c}(S^{'})\$‎, ‎where‎ ‎\$eta(S)\$‎, ‎\$L(S)\$‎, ‎\$J(S)\$ and \$T_{1}(S)\$ are negation‎, ‎line graph‎, ‎jump graph and semitotal line sigraph of \$S\$‎, ‎respectively‎, ‎and‎ ‎\$sim\$ means switching equivalence‎. signed graph Line sigraph Jump sigraph Semitotal line sigraph Lict sigraph 2014 12 01 11 18 http://toc.ui.ac.ir/article_5627_e7de2aef7c26e21d97bfaf79f2112406.pdf Behav‎. ‎Sci. R‎. ‎P‎. ‎Abelson and M‎. ‎J‎. ‎Rosenberg 3 1 1958 J‎. ‎Combin‎. ‎Math‎. ‎Combin‎. ‎Comput. M‎. ‎Acharya 69 103 2009 ‎Graph Theory Notes of New York M‎. ‎Acharya and D‎. ‎Sinha XLIV 30 2003 J‎. ‎Math‎. ‎Psychol. L‎. ‎W‎. ‎Beineke and F‎. ‎Harary 18 260 1978 Acta Cienc‎. ‎Indica Math. B‎. ‎Basavanagoud and Veena N‎. ‎Mathad 31 735 2005 ‎Discrete Math. G‎. ‎Chartrand‎, ‎H‎. ‎Hevia‎, ‎E‎. ‎B‎. ‎Jarrett and M‎. ‎Schultz 170 63 1997 Int‎. ‎J‎. ‎Contemp‎. ‎Math‎. ‎Sci. D‎. ‎Sinha and P‎. ‎Garg 6 221 2011 ‎Addison-Wesley Publishing Co.‎, ‎Reading‎, ‎Mass.-Menlo Park‎, ‎Calif.-London F‎. ‎Harary 1969 Michigan Math‎. ‎J. F‎. ‎Harary 2 143 1953-54 Behavioral Sci. F‎. ‎Harary 2 255 1957 J‎. ‎of Analysis and Comput. V‎. ‎R‎. ‎Kulli and M‎. ‎H‎. ‎Muddebihal 2 33 2006 Nat‎. ‎Acad‎. ‎Sci‎. ‎Lett. E‎. ‎Sampathkumar 7 91 1984 J‎. ‎Karnatak Univ‎. ‎Sci. E‎. ‎Sampathkumar and S‎. ‎B‎. ‎Chikkodimath 18 281 1973 J‎. ‎Graph Theory T‎. ‎Sozansky 4 127 1980 Electronic J‎. ‎Combin. T‎. ‎Zaslavski 8 1998 Discrete Appl‎. ‎Math. T‎. ‎Zaslavski 4 47 1982
2014-12-01 10.22108
Transactions on Combinatorics Trans. Comb. 2251-8657 2251-8657 2014 3 4 The geodetic domination number for the product of graphs S. Robinson Chellathurai S. Padma Vijaya A subset \$S\$ of vertices in a graph \$G\$ is called a geodetic set if every vertex not in \$S\$ lies on a shortest path between two vertices from \$S\$‎. ‎A subset \$D\$ of vertices in \$G\$ is called dominating set if every vertex not in \$D\$ has at least one neighbor in \$D\$‎. ‎A geodetic dominating set \$S\$ is both a geodetic and a dominating set‎. ‎The geodetic (domination‎, ‎geodetic domination) number \$g(G) (gamma(G),gamma_g(G))\$ of \$G\$ is the minimum cardinality among all geodetic (dominating‎, ‎geodetic dominating) sets in \$G\$‎. ‎In this paper‎, ‎we show that if a triangle free graph \$G\$ has minimum degree at least 2 and \$g(G) = 2\$‎, ‎then \$gamma _g(G) = gamma(G)\$‎. ‎It is shown‎, ‎for every nontrivial connected graph \$G\$ with \$gamma(G) = 2\$ and \$diam(G) > 3\$‎, ‎that \$gamma_g(G) > g(G)\$‎. ‎The lower bound for the geodetic domination number of Cartesian product graphs is proved‎. ‎Geodetic domination number of product of cycles (paths) are determined‎. In this work‎, ‎we also determine some bounds and exact values of the geodetic domination number of strong product of graphs‎. Cartesian product strong product geodetic number Domination Number geodetic domination number 2014 12 01 19 30 http://toc.ui.ac.ir/article_5750_87bdcf395fa6e3fd7e39a154bc0f1442.pdf Discrete Math. B. Bresar, S. Klavzar and A. T. Horvat 308 5555 2008 Addison-Wesley Publishing Company, Redwo o d City, CA, F. Buckley and F. Harary 1990 Comput. Math. Appl. J. Caceres, C. Hernando, M. Mora and I. M. Pelayo 60 3020 2010 J. Combin. Math. Combin. Comput. H. Escuadro, R. Gera, A. Hansb erg, N. Jafari Rad and L. Volkmann 77 89 2011 Networks G. Chartrand, F. Harary and P. Zhang 39 1 2002 Discrete Math. A. Hansb erg and L. Volkmann 310 2140 2010 Math. Comput. Model ling F. Harary, E. Loukakis and C. Tsouros 17 89 1993 Marcel Dekker, Inc, New York T. W. Haynes, S. T. Hedetniemi and P. J. Slater 208 1998 Ars Combin. M. S. Jacobson and L. F. Kinch 18 33 1984 IEEE J. Cao, B. Wu and M. Shi 2009 Discrete Appl. Math. S. Klavzar and N. Seifter 59 129 1995 Discuss. Math. Graph Theory A. P. Santhakumaran and S. V. Ullas Chandran 30 687 2010 T. Jiang, I. Pelayo and D. Pritikin 2004 Manuscript I. G. Yero and J. A. Ro driguez-velazquez Applicable Analysis and Discrete Mathematics I. G. Yero and J. A. Ro driguez-velazquez 7 262 2013
2014-12-01 10.22108
Transactions on Combinatorics Trans. Comb. 2251-8657 2251-8657 2014 3 4 Comparing the second multiplicative Zagreb coindex with some graph invariants Farzaneh Falahati Nezhad Ali Iranmanesh Abolfazl Tehranian Mahdieh Azari ‎‎The second multiplicative Zagreb coindex of a simple graph \$G\$ is‎ ‎defined as‎: ‎\$\${overline{Pi{}}}_2left(Gright)=prod_{uvnotin{}E(G)}d_Gleft(uright)d_Gleft(vright),\$\$‎ ‎where \$d_Gleft(uright)\$ denotes the degree of the vertex \$u\$ of‎ ‎\$G\$‎. ‎In this paper‎, ‎we compare \$overline{{Pi}}_2\$-index with‎ ‎some well-known graph invariants such as the Wiener index‎, ‎Schultz‎ ‎index‎, ‎eccentric connectivity index‎, ‎total eccentricity‎, ‎eccentric-distance sum‎, ‎the first Zagreb index and coindex and the‎ ‎first multiplicative Zagreb index and coindex‎. Degree (in graphs) Topological index multiplicative Zagreb coindex 2014 12 01 31 41 http://toc.ui.ac.ir/article_5951_503474fbf2c1d8a206d3e6dd20f8a32e.pdf J‎. ‎Comput‎. ‎Theor‎. ‎Nanosci. Y‎. ‎Alizadeh‎, ‎M‎. ‎Azari and T‎. ‎Doslic 10 1297 2013 Discrete Appl‎. ‎Math. A‎. ‎R‎. ‎Ashrafi‎, ‎T‎. ‎Dos}lic and A‎. ‎Hamzeh 158 1571 2010 Appl‎. ‎Math‎. ‎Comput. ‎Azari‎ 239 409 2014 MATCH Commun‎. ‎Math‎. ‎Comput‎. ‎Chem. M‎. ‎Azari and A‎. ‎Iranmanesh 70 901 2013 Discrete Appl‎. ‎Math. M‎. ‎Azari and A‎. ‎Iranmanesh 161 2827 2013 J‎. ‎Math‎. ‎Inequal., ‎(to appear) M‎. ‎Azari and A‎. ‎Iranmanesh Util‎. ‎Math. P‎. ‎Dankelmann‎, ‎W‎. ‎Goddard and C‎. ‎S‎. ‎Swart 65 41 2004 Trans‎. ‎Comb. M‎. ‎Eliasi 1 17 2012 MATCH Commun‎. ‎Math‎. ‎Comput‎. ‎Chem. M‎. ‎Eliasi‎, ‎A‎. ‎Iranmanesh and I‎. ‎Gutman 68 217 2012 MATCH Commun‎. ‎Math‎. ‎Comput‎. ‎Chem. M‎. ‎Eliasi and D‎. ‎Vukicevic 69 765 2013 Ars Combin., ‎(to appear) F‎. ‎Falahati Nezhad‎, ‎A‎. ‎Iranmanesh‎, ‎A‎. ‎Tehranian and M‎. ‎Azari J‎. ‎Math‎. ‎Anal‎. ‎Appl. S‎. ‎Gupta‎, ‎M‎. ‎Singh and A‎. ‎K‎. ‎Madan 275 386 2002 ‎Bull‎. ‎Int‎. ‎Math‎. ‎Virtual Inst. I‎. ‎Gutman 1 13 2011 pringer‎, ‎Berlin I‎. ‎Gutman and O‎. ‎E‎. ‎Polansky 1986 Chem‎. ‎Phys‎. ‎Lett. I‎. ‎Gutman and N‎. ‎Trinajstic 17 535 1972 J‎. ‎Math‎. ‎Anal‎. ‎Appl. A‎. ‎Ilic‎, ‎G‎. ‎Yu and L‎. ‎Feng 381 590 2011 Discrete‎ ‎Appl‎. ‎Math. M‎. ‎H‎. ‎Khalifeh‎, ‎H‎. ‎Yusefi-Azari and A‎. ‎R‎. ‎Ashrafi 157 804 2009 Croat‎. ‎Chem‎. ‎Acta. S‎. ‎Nikolic‎, ‎G‎. ‎Kovacevic‎, ‎A‎. ‎Milicevic and N‎. ‎Trinajstic 76 113 2003 Bull‎. ‎Int‎. ‎Math‎. ‎Virtual Inst. T‎. ‎Reti and I‎. ‎Gutman 2 133 2012 J‎. ‎Chem‎. ‎Inf‎. ‎Comput‎. ‎Sci. H‎. ‎P‎. ‎Schultz 29 227 1989 J‎. ‎Chem‎. ‎Inf‎. ‎Comput‎. ‎Sci. V‎. ‎Sharma‎, ‎R‎. ‎Goswami and A‎. ‎K‎. ‎Madan 37 273 1997 MATCH Commun‎. ‎Math‎. ‎Comput‎. ‎Chem. R‎. ‎Todeschini and V‎. ‎Consonni 64 359 2010 J‎. ‎Amer‎. ‎Chem‎. ‎Soc. H‎. ‎Wiener 69 2636 1947 J‎. ‎Am‎. ‎Chem‎. ‎Soc. H‎. ‎Wiener 69 17 1947 Opuscula Math. K‎. ‎Xu‎, ‎K‎. ‎C‎. ‎Das and K‎. ‎Tang 33 191 2013 MATCH Commun‎. ‎Math‎. ‎Comput‎. ‎Chem. K‎. ‎Xu and H‎. ‎Hua 68 241 2012 MATCH Commun‎. ‎Math‎. ‎Comput‎. ‎Chem. B‎. ‎Zhou 52 113 2004
2014-12-01 10.22108
Transactions on Combinatorics Trans. Comb. 2251-8657 2251-8657 2014 3 4 Perfect state transfer in unitary Cayley graphs over local rings Yotsanan Meemark Songpon Sriwongsa ‎In this work‎, ‎using eigenvalues and eigenvectors of unitary Cayley graphs over finite local rings and elementary linear algebra‎, ‎we characterize which local rings allowing PST occurring in its unitary Cayley graph‎. ‎Moreover‎, ‎we have some developments when \$R\$ is a product of local rings‎. Local rings Perfect state transfer, Unitary Cayley graphs 2014 12 01 43 54 http://toc.ui.ac.ir/article_5974_ee7986b6514f7db7db00ebf91f7de927.pdf Electron. J. Combin. R. Akhtar, M. Boggess, T. Jackson-Henderson, I. Jimenez, R. Karpman, A. Kinzel and D. Pritikin 16 13 2009 Quantum Inf. Comput. R. J. Angeles-Canul, R. Norton, M. Opperman, C. Paribello, M. Russell and C. Tamon 10 325 2010 Appl. Math. Lett. M. Basic, M. D. Petkovic and D. Stevanovic 22 1117 2009 Linear Al gebra Appl. W.-C. Cheung and C. Godsil 435 2468 2011 4th edn, Prentice Hall, New Jersey S. H. Friedberg, A. J. Insel and L. E. Spence 2003 Math. Ann. N. Ganesan 157 215 1964 Int. J. Quantum Inf. Y. Ge, B. Greenberg, O. Perez and C. Tamon 9 823 2011 Discrete Math. C. Godsil 312 129 2012 Contemporary Physics J. Kempe 44 307 2003 Int. J. Quantum Inform. V. Kendon 4 791 2006 Linear Algebra Appl. D. Kiani, M. M. H. Aghaei, Y. Meemark and B. Suntornpoch 435 1336 2011 Amer. Math. Monthly D. Singmaster and D. M. Bloom 71 918 1964
2014-12-01 10.22108
Transactions on Combinatorics Trans. Comb. 2251-8657 2251-8657 2014 3 4 Complete solution to a conjecture of Zhang-Liu-Zhou Mostafa Tavakoli F. Rahbarnia M. Mirzavaziri A. R. Ashrafi ‎‎Let \$d_{n,m}=big[frac{2n+1-sqrt{17+8(m-n)}}{2}big]\$ and‎ ‎\$E_{n,m}\$ be the graph obtained from a path‎ ‎\$P_{d_{n,m}+1}=v_0v_1 cdots v_{d_{n,m}}\$ by joining each vertex of‎ ‎\$K_{n-d_{n,m}-1}\$ to \$v_{d_{n,m}}\$ and \$v_{d_{n,m}-1}\$‎, ‎and by‎ ‎joining \$m-n+1-{n-d_{n,m}choose 2}\$ vertices of \$K_{n-d_{n,m}-1}\$‎ ‎to \$v_{d_{n,m}-2}\$‎. ‎Zhang‎, ‎Liu and Zhou [On the maximal eccentric‎ ‎connectivity indices of graphs‎, ‎Appl‎. ‎Math‎. ‎J‎. ‎Chinese Univ.‎, ‎in‎ ‎press] conjectured that if \$d_{n,m}geqslant 3\$‎, ‎then \$E_{n,m}\$‎ ‎is the graph with maximal eccentric connectivity index among all‎ ‎connected graph with \$n\$ vertices and \$m\$ edges‎. ‎In this note‎, ‎we‎ ‎prove this conjecture‎. ‎Moreover‎, ‎we present the graph with‎ ‎maximal eccentric connectivity index among the connected graphs‎ ‎with \$n\$ vertices‎. ‎Finally‎, ‎the minimum of this graph invariant‎ ‎in the classes of tricyclic and tetracyclic graphs are computed‎. Eccentric connectivity index tricyclic graph tetracyclic graph graph operation 2014 12 01 55 58 http://toc.ui.ac.ir/article_5986_740f215bc6659e95ccaa77c44e50e504.pdf Prentice Hall, Inc., Upper Saddle River, NJ D. B. West 1996 J. Chem. Inf. Comput. Sci. V. Sharma, R. Goswami and A. K. Madan 37 273 1997 MATCH Commun. Math. Comput. Chem. A. R. Ashrafi, T. Doslic and M. Saheli 65 221 2011 J. Comput. Appl. Math. A. R. Ashrafi, M. Saheli and M. Ghorbani 235 4561 2011 J. Math. Anal. Appl. G. Yu, L. Feng and A. Ilic 375 99 2011 MATCH Commun. Math. Comput. Chem. B. Zhou and Z. Du 63 181 2010 MATCH Commun. Math. Comput. Chem. A. Ilic and I. Gutman 65 731 2011 Discrete Math. M. J. Morgan, S. Mukwembi and H. C. Swart 311 1229 2011 Iranian J. Math. Chem. T. Doslic, M. Saheli and D. Vukicevic 1 45 2010 Appl. Math. J. Chinese Univ. J. Zhang, Z. Liu and B. Zhou in press