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The common minimal common neighborhood dominating signed graphs
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In this paper, we define the common minimal common neighborhood dominating signed graph (or common minimal $CN$dominating signed graph) of a given signed graph and offer a structural characterization of common minimal $CN$dominating signed graphs. In the sequel, we also obtained switching equivalence
characterization: $overline{Sigma} sim CMCN(Sigma)$, where $overline{Sigma}$ and $CMCN(Sigma)$ are complementary signed graph and common minimal $CN$signed graph of $Sigma$ respectively.
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P. Siva
Reddy
Dept. of Mathematics, Acharya Institute of Technology, Bangalore560 090, India.
Dept. of Mathematics, Acharya Institute of
India
reddy_math@yahoo.com


K. R.
Rajanna
Professor and Head
Dept. of Mathematics
Acharya Institute of Technology
Bangalore560 090
India
Professor and Head
Dept. of Mathematics
Acharya
India
rajanna@acharya.ac.in


Kavita
Permi
Assistant Professor
Dept.of Mathematics
Acharya Institute of Technology
Bangalore560 090
India.
Assistant Professor
Dept.of Mathematics
Acharya
India
kavithapermi@acharya.ac.in
Signed graphs
Balance
Switching
Common minimal $CN$dominating signed graph
Negation
[R. P. Abelson and M. J. Rosenberg (1958). Symoblic psychologic: A model of
attitudinal cognition. Behav. Sci.. 3, 113## A. Alwardi, N. D. Soner and K. Ebadi (2011). On the common neighbourhood domination
number. J. Comp. $&$ Math. Sci.. 2 (3) , 547556## A. Alwardi and N. D. Soner (2012). Minimal, vertex minimal and commonality minimal CNdominating
graphs. Trans. Comb.. 1 (1) , 2129## C. Berge (1962). Theory of Graphs and its Applications. Methuen, London. ## E. J. Cockayne and S. T. Hedetniemi (1977). Towards a theory of domination in graphs. Networks. 7, 247261## C. F. De Jaenisch (1862). Applications de l’Analyse mathematique an Jen des Echecs. ## D. Easley and J. Kleinberg (2010). Networks, Crowds, and Markets: Reasoning About
a Highly Connected World. Cambridge University Press. ## F. Harary (1969). Graph Theory. AddisonWesley Publishing Co.. ## F. Harary (1953). On the notion of balance of a signed graph. Michigan Math. J.. 2, 143146## F. Harary (1957). Structural duality. Behavioral Sci.. 2 (4) , 255265## O. Ore (1962). Theory of Graphs. Amer. Math. Soc. Colloq. Publ.. 38## R. Rangarajan and P. Siva Kota Reddy (2010). The edge $C_4$ signed graph of a signed graph. Southeast Asian Bull. Math.. 34 (6) , 10771082## R. Rangarajan, M. S. Subramanya and P. Siva Kota Reddy (2012). Neighborhood signed graphs. Southeast Asian Bull. Math.. 36 (3) , 389397## W. W. Rouse Ball (1892). Mathematical Recreation and Problems of Past and Present Times. ## E. Sampathkumar (1984). Point signed and line signed graphs. Nat.
Nat. Acad. Sci. Lett.. 7 (3) , 9193## E. Sampathkumar, P. Siva Kota Reddy and M. S.
Subramanya (2010). Directionally $n$signed graphs. Ramanujan Math. Soc., Lecture Notes
Series (Proc. Int. Conf. ICDM 2008). 13, 155162## E. Sampathkumar, P. Siva Kota Reddy and M. S. Subramanya (2009). Directionally $n$signed graphsII. Int. J. Math. Comb.. 4, 8998## E. Sampathkumar, M. S. Subramanya and P. Siva Kota Reddy (2011). Characterization of line sidigraphs. Southeast Asian Bull. Math.. 35 (2) , 297304## P. Siva Kota Reddy and M. S. Subramanya (2009). Note on path signed graphs. Notes on Number Theory and Discrete Mathematics. 15 (4) , 16## P. Siva Kota Reddy, S. Vijay and V. Lokesha (2009). $n^{th}$ Power signed graphs. Proc. Jangjeon Math. Soc.. 12 (3) , 307313## P. Siva Kota Reddy (2010). $t$Path Sigraphs. Tamsui Oxf. J. Math. Sci.. 26 (4) , 433441## P. Siva Kota Reddy, E. Sampathkumar and M. S. Subramanya (2010). Commonedge signed graph of a
signed graph. J. Indones. Math. Soc.. 16 (2) , 105112## P. Siva Kota Reddy, B. Prashanth and T. R. Vasanth Kumar (2011). Antipodal signed directed
Graphs. Advn. Stud. Contemp. Math.. 21 (4) , 355360## P. Siva Kota Reddy and B. Prashanth (2012). The Common Minimal Dominating Signed
Graph. Trans. Comb.. 1 (3) , 3946## P. Siva Kota Reddy and B. Prashanth (2012). $mathcal{S}$Antipodal signed graphs. Tamsui Oxford J. of Inf. Math. Sciences. 28 (2) , 165174## P. Siva Kota Reddy and S. Vijay (2012). The super line signed graph $mathcal{L}_r(S)$ of a signed Graph. Southeast Asian Bulletin of Mathematics. 36 (6) , 875882## P. Siva Kota Reddy and U. K. Misra (2012). Common Minimal Equitable Dominating Signed
Graphs. Notes on Number Theory and Discrete Mathematics. 18 (4) , 4046## P. Siva Kota Reddy and U. K. Misra (2013). The Equitable Associate Signed
Graphs. Bull. Int. Math. Virtual Inst.. 3 (1) , 1520## P. Siva Kota Reddy, U. K. Misra and P. N. Samanta (2013). The Minimal Equitable Dominating Signed
Graphs. Bull. of Pure $&$ Appl. Math., to appear. ## P. Siva Kota Reddy and B. Prashanth (2012). Note on Minimal Dominating Signed
Graphs. Bull. of Pure $&$ Appl. Math., to appear. ## T. Soz$acute{a}$nsky (1980). Enueration of weak isomorphism
classes of signed graphs. J. Graph Theory. 4 (2) , 127144## A. M. Yaglom and I. M. Yaglom (1964). Challenging mathematical problems with elementary solutions. Combinatorial Analysis and Probability Theory. 1## T. Zaslavsky (1982). Signed graphs. Discrete Appl. Math.. 4 (1) , 4774## T. Zaslavsky (1998). A mathematical bibliography of signed and gain graphs and its allied areas. Electron. J. Combin., Dynamic Surveys, no. DS8. 8 (1) ## ]
Bounding the domination number of a tree in terms of its annihilation number
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A set $S$ of vertices in a graph $G$ is a dominating set if every vertex of $VS$ is adjacent to some vertex in $S$. The domination number $gamma(G)$ is the minimum cardinality of a dominating set in $G$. The annihilation number $a(G)$ is the largest integer $k$ such that the sum of the first $k$ terms of the nondecreasing degree sequence of $G$ is at most the number of edges in $G$. In this paper, we show that for any tree $T$ of order $nge 2$, $gamma(T)le frac{3a(T)+2}{4}$, and we characterize the trees achieving this bound.
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Nasrin
Dehgardai
Azarbaijan Shahid Madani University
Azarbaijan Shahid Madani University
Iran
ndehgardi@gmail.com


Sepideh
Norouzian
Azarbaijan Shahid Madani University
Azarbaijan Shahid Madani University
Iran
s_maleki494@yahoo.com


Seyed Mahmoud
Sheikholeslami
Azarbaijan University of Tarbiat Moallem
Azarbaijan University of Tarbiat Moallem
Iran
s.m.sheikholeslami@azaruniv.edu
annihilation number
dominating set
Domination Number
[H. Aram, A. Bahremandpour, A. Khodkar, S. M. Sheikholeslami and
L. Volkmann Relating the annihilation number and the Roman
domination number of a tree. submitted. ## Y. Caro and R. Pepper (2012). Degree Sequence Index Strategy. arXiv:1210.1805v1. ## E. J. Cockayne, C. W. Ko and F. B. Shepherd (1985). Inequalities concerning dominating sets in
graphs. Technical Report DM370IR, Dept. Math. Univ. Victoria. ## E. DeLaVina, C. E. Larson, R. Pepper and B. Waller (2010). Graffiti.pc on the 2domination number of a graph. Congr. Numer.. 203, 1532## E. DeLaVina, R. Pepper and B. Waller (2007). Independence, radius,
and Hamiltonian paths. MATCH Commun. Math. Comput. Chem.. 58, 481510## W. J. Desormeaux, T. W. Haynes and M. A. Henning (2013). Relating the annihilation number and the total domination number
of a tree. Discrete Appl. Math.. 161, 349354## T. W. Haynes, S. T. Hedetniemi and P. J. Slater (1998). Fundamentals of Domination in Graphs. Marcel Dekker, Inc.,
New York. ## T. W. Haynes, S. T. Hedetniemi, and P. J. Slater (1998). Domination in Graphs: Advanced Topics. Marcel Dekker, Inc.,
New York. ## L. Jennings (2008). New Sufficient Condition for Hamiltonian Paths. Ph.D. Dissertation, Rice University. ## C. E. Larson and R. Pepper (2011). Graphs with equal independence
and annihilation numbers. The Electron. J. Combin., $#$P180. 18## O. Ore (1962). Theory of Graphs. Amer. Math. Soc. Colloq. Publ., (Amer. Math. Soc., Providence, RI). 38## R. Pepper (2004). Binding Independence. Ph.D. Dissertation,
University of Houston. ## R. Pepper (2009). On the annihilation number of a graph. Recent Advances In Electrical
Engineering: Proceedings of the 15th American Conference on
Applied Mathematics. , 217220## B. Reed (1996). Paths, stars and the number three. Combin. Probab. Comput.. 5, 277295## D. B. West (1996). Introduction to Graph Theory. Prentice Hall, Inc., Upper Saddle River, NJ. ## ]
Gray isometries for finite $p$groups
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We construct two classes of Gray maps, called typeI Gray map and typeII Gray map, for a finite $p$group $G$. TypeI Gray maps are constructed based on the existence of a Gray map for a maximal subgroup $H$ of $G$. When $G$ is a semidirect product of two finite $p$groups $H$ and $K$, both $H$ and $K$ admit Gray maps and the corresponding homomorphism $psi:Hlongrightarrow {rm Aut}(K)$ is compatible with the Gray map of $K$ in a sense which we will explain, we construct typeII Gray maps for $G$. Finally, we consider group codes over the dihedral group $D_8$ of order 8 given by the set of their generators, and derive a representation and an encoding procedure for such codes.
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Reza
Sobhani
Iran
r.sobhani@sci.ui.ac.ir
Finite group
Code
Gray map
Isometry
[A. A. Nechaev (1991). Kerdock code in a cyclic form. Discrete Math. Appl.. 1, 365384## A. R. Hammons, P. V. Kummar, A. R. Calderbank, N. J. A. Sloane and P. Sole (1994). The ${mathbb Z}_4$
linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inform. Theory. 40, 301319## A. R. Calderbank and G. McGuire (1997). Construction of a $(64,2^{37},12)$ code via Galois rings. Des. Codes Cryptogr.. 10, 157165## M. Greferath and S. E. Schmidt (1999). Gray isometries for finite chain rings and a nonlinear ternary $(36,3^{12},15)$code. IEEE Trans. Inform. Theory. 45, 25222524## I. M. Duursma, M. Greferath, S. Litsyn, and S. E. Schmidt (2001). A ${mathbb Z}_8$linear lift of the binary Golay code and a
nonlinear binary $(96, 2^{37}, 24)$code. IEEE Trans. Inform. Theory. 47, 15961598## M. Kiermaier and J. Zwanzger (2011). A ${mathbb Z}_4$linear code of high minimum Lee distance derived from
a hyperoval. Adv. Math. Commun.. 5, 275286## I. Constantinescu and W. Heise (1997). A metric for codes over residue class rings. Problems Inform. Transmission. 33, 208213## H. TapiaRecillas and G. Vega (2003). Some constacyclic codes over ${mathbb Z}_{2^k}$ and binary quasicyclic
codes. Discrete Appl. Math.. 128, 305316## S. Ling and T. Blackford (2002). ${mathbb Z}_{p^{k+1}}$linear codes. IEEE Trans. Inform. Theory. 48, 25922605## J. F. Qian, L. N. Zhang and S. X. Zhu (2006). $(1+u)$ Constacyclic and cyclic codes over ${mathbb F}_2+u{mathbb F}_2$. Appl. Math. Lett.. 19, 820823## J. F. Qian, L. N. Zhang and S. X. Zhu (2006). Constacyclic and cyclic codes over ${mathbb F}_2+u{mathbb F}_2+u^2{mathbb F}_2$. IEICE Trans. Fundamentals. E89A, 18631865## R. Sobhani and M. Esmaeili (2010). Some Constacyclic and cyclic codes over ${mathbb F}_q[u]/left$. IEICE Trans. Fundamentals. E93A, 808813## G. D. Forney (1992). On the Hamming distance properties of group codes. IEEE Trans. Inform. Theory. 38, 17971801## ]
New skew Laplacian energy of simple digraphs
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For a simple digraph $G$ of order $n$ with vertex set ${v_1,v_2,ldots, v_n}$, let $d_i^+$ and $d_i^$ denote the outdegree and indegree of a vertex $v_i$ in $G$, respectively. Let $D^+(G)=diag(d_1^+,d_2^+,ldots,d_n^+)$ and $D^(G)=diag(d_1^,d_2^,ldots,d_n^)$. In this paper we introduce $widetilde{SL}(G)=widetilde{D}(G)S(G)$ to be a new kind of skew Laplacian matrix of $G$, where $widetilde{D}(G)=D^+(G)D^(G)$ and $S(G)$ is the skewadjacency matrix of $G$, and from which we define the skew Laplacian energy $SLE(G)$ of $G$ as the sum of the norms of all the eigenvalues of $widetilde{SL}(G)$. Some lower and upper bounds of the new skew Laplacian energy are derived and the digraphs attaining these bounds are also determined.
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Qingqiong
Cai
Center for Combinatorics, nankai University, Tianjin, China
Center for Combinatorics, nankai University,
China
cqqnjnu620@163.com


Xueliang
Li
Center for Combinatorics, Nankai University, Tianjin 300071, China
Center for Combinatorics, Nankai University,
China
lxl@nankai.edu.cn


Jiangli
Song
Center for Combinatorics, Nankai University, Tianjin, China
Center for Combinatorics, Nankai University,
China
songjiangli@mail.nankai.edu.cn
energy
Laplacian energy
skew energy
skew Laplacian energy
eigenvalues
[C. Adiga, R. Balakrishnan and W. So (2010). The shew energy
of a digraph. Linear Algebra Appl.. 432, 18251835## C. Adiga and M. Smitha (2009). On the skew Laplacian energy of a digraph. Int. Math. Forum. 4 (3) , 19071914## C. Adiga and Z. Khoshbakht (2009). On some inequalities for the skew Laplacian energgy of digraphs. JIPAM. J. Inequal. Pure Appl. Math.. 10 (3) , 6## D. Cvetkovi$acute{c}$, P. Rowlinson and
S. Simi$acute{c}$ (2010). An Introduction to the Theory of Graph
Spectra. Cambridge Univ. Press, Cambridge. ## I. Gutman (1978). The energy of a graph. Ber. Math.Statist. Sekt. Forsch. Graz. 103, 122## I. Gutman and B. Zhou (2006). Laplacian energy of
a graph. Linear Algebra Appl.. 414, 2937## I. Gutman, X. Li and J. Zhang (2009). Graph Energy,
in: M. Dehmer, F. EmmertStreib (Eds.). Analysis of Complex
Network: From Biology to Linguistics, WileyVCH Verlag, Weinheim. , 145174## R. A. Horn and C. R. Johnson (1990). Matrix Analysis. Cambridge Univ. Press. ## M. L. Kragujevac (2006). On the Laplacian energy
of a graph. Czech. Math. J.. 56 (131) , 12071213## X. Li, Y. Shi and I. Gutman (2012). Graph Energy. Springer, New York. ## P. Kissani and Y. Mizoguchi (2010). Laplacian energy of directed graphs and minimizing maximum outdegree algorithms. Kyushu University Institutional Repository. ## ]
A comprehensive survey: Applications of multiobjective particle swarm optimization (MOPSO) algorithm
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Numerous problems encountered in real life cannot be actually formulated as a single objective problem; hence the requirement of MultiObjective Optimization (MOO) had arisen several years ago. Due to the complexities in such type of problems powerful heuristic techniques were needed, which has been strongly satisfied by Swarm Intelligence (SI) techniques. Particle Swarm Optimization (PSO) has been established in 1995 and became a very mature and most popular domain in SI. MultiObjective PSO (MOPSO) established in 1999, has become an emerging field for solving MOOs with a large number of extensive literature, software, variants, codes and applications. This paper reviews all the applications of MOPSO in miscellaneous areas followed by the study on MOPSO variants in our next publication. An introduction to the key concepts in MOO is followed by the main body of review containing survey of existing work, organized by application area along with their multiple objectives, variants and further categorized variants.
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Soniya
Lalwani
Statistician, R & D, Advanced Bioinformatics Centre, Birla Institute of Scientific Research, Jaipur
PhD student, Department of Mathematics, Malaviya National Institute of Technology, Jaipur
Statistician, R & D, Advanced Bioinformatics
India
slalwani.math@gmail.com


Sorabh
Singhal
Project student, R & D, Advanced Bioinformatics Centre, Birla Institute of Scientific Research, Jaipur
Project student, R & D, Advanced Bioinformatics
India
saurabhez@gmail.com


Rajesh
Kumar
Associate Professor, Department of Electrical Engineering, Malaviya National Institute of Technology, Jaipur
Associate Professor, Department of Electrical
India
rkumar.ee@gmail.com


Nilama
Gupta
Associate Professor, Department of Mathematics, Malaviya National Institute of Technology, Jaipur
Associate Professor, Department of Mathematics,
India
guptanilama@gmail.com
MultiObjective Particle Swarm Optimization
Conflicting objectives
Particle swarm optimization
Pareto optimal set
Nondominated solutions
[Abido (2007). Multiobjective particle swarm for environmental/economic dispatch problem. International Power Engineering Conference. , 13851390## M. A. Abido (2010). Multiobjective particle swarm optimization for optimal power flow problem. Handbook of Swarm Intelligence, adaptation, learning and optimization, Springer. , 241268## S. Agrawal, B. K. Panigrahi and M. J. Tiwari (2008). Multiobjective particle swarm algorithm with fuzzy clustering for electrical power dispatch. IEEE Transactions on Evolutionary Computation. 12 (5) , 529541## A. Ajami and M. Armaghan (2010). Application of multiobjective PSO algorithm for power system stability enhancement by means of SSSC. International Journal of Computer and Electrical Engineering. 2 (5) , 838845## B. Alatas and E. Akin (2009). Multiobjective rule mining using a chaotic particle swarm optimization algorithm. Knowledge Based Systems. 22 (6) , 455460## ]
Eccentric connectivity index and eccentric distance sum of some graph operations
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Let $G=(V,E)$ be a connected graph. The eccentric connectivity index of $G$, $xi^{c}(G)$, is defined as
$xi^{c}(G)=sum_{vin V(G)}deg(v)ec(v)$, where $deg(v)$ is the degree of a vertex $v$ and $ec(v)$ is its eccentricity. The eccentric distance sum of $G$ is defined as $xi^{d}(G)=sum_{vin V(G)}ec(v)D(v)$, where $D(v)=sum_{uin V(G)}d(u,v)$. In this paper, we calculate the eccentric connectivity index and eccentric distance sum of generalized hierarchical product of graphs. Moreover, we present the exact formulae for the eccentric connectivity index of $F$sum graphs in terms of some invariants of the factors.
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111


Buzohragul
Eskender
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, P.R. China
College of Mathematics and System Sciences,
China
buzoragul2005@163.com


Elkin
Vumar
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China
College of Mathematics and System Sciences,
China
vumar@xju.edu.cn
Eccentric connectivity index
eccentric distance sum
generalized hierarchical product
$F$sum graphs
[A. R. Ashrafi, M. Saheli and M. Ghorbani (2011). The eccentric connectivity index of nanotubes and nanotori. J. Comput. Appl. Math.. 235, 45614566## L. Barri`{e}re, F. Comellas, C. Dalf'{o} and M. A. Fiol (2009). The hierarchical product of graphs. Discrete Appl. Math.. 157, 3648## L. Barri`{e}re, C. Dalf'{o}, M. A. Fiol and M. Mitjana (2009). The generalized hierarchical product of graphs. Discrete Math.. 309, 38713881## J. A. Bondy and U. S. R. Murty (2008). Graph Theory. Graduate Texts in Mathematics, Springer, New York. 244## P. Dankelmann, W. Goddard and C. S. Swart (2004). The average eccentricity of a graph and its subgraphs. Util. Math.. 65, 4151## R. Entringer and I. Gutman (2001). Wiener index of trees: theory and applications. Acta Appl. Math.. 66, 211249## T. Dov{s}li'{c}, A. Graovac and O. Ori (2011). Eccentric connectivity index of hexagonal belts and chains. MATCH Commun. Math. Compu. Chem.. 65, 745752## M. Eliasi and A. Iranmanesh (2011). The hyperWiener index of the generalized hierarchical product of graphs. Discrete Appl. Math.. 159, 866871## M. Eliasi and B. Taeri (2009). Four new sums of graphs and their Wiener indices. Discrete Appl. Math.. 157, 794803## S. Gupta, M. Singh and A. K. Madan (2002). Application of graph theory: Relationship of eccentric connectivity index and Wiener's index with antiinflammatory activity. J. Math. Anal. Appl.. 266, 259268## H. Hua, S. Zhang and K. Xu (2012). Further results on the eccentric distance sum. Discrete Appl. Math.. 160, 170180## A. Ili'{c} and I. Gutman (2011). Eccentric connectivity index of chemical trees. MATCH Commun. Math. Comput. Chem.. 65, 731744## A. Ili'{c} (2010). Eccentric connectivity index, in: I. Gutman, B. Furtula (Eds.), Novel Molecular Structure
Descriptors  Theory and Applications II. Math. Chem. Monogr., University of Kragujevac. 9, 139168## A. Ili'{c}, G. Yu and L. Feng (2011). On the eccentric distance sum of graphs. J. Math. Anal. Appl.. 381, 590600## S. Li and G. Wang (2011). Vertex PI indices of four sums of graphs. Discrete Appl. Math.. 159, 16011607## M. Metsidik, W. Zhang and F. Duan (2010). Hyper and reverse Wiener indices of $F$sums of graphs. Discrete Appl. Math.. 158, 14331440## M. J. Morgan, S. Mukwembi and H. C. Swart (2011). On the eccentric connectivity index of a graph. Discrete Math.. 311, 12291234## M. H. Khalifeh, H. YousefiAzari and A. R. Ashrafi (2008). The hyperWiener index of graph operations. Comput. Math. Appl.. 56, 14021407## S. Sardana and A. K. Madan (2001). Application of graph theory: Relationship of molecular connectivity index, Wiener's index
and eccentric connectivity index with diuretic activity. MATCH Commun. Math. Compute. Chem.. 43, 8598## V. Sharma, R. Goswami and A. K. Madan (1997). Eccentric connectivity index: A novel highly discriminating topological descriptor for structureproperty and structureactivity studies. J. Chem. Inform. Model.. 37, 273282## G. Yu, L. Feng and A. Ili'{c} (2011). On the eccentric distance sum of trees and unicyclic graphs. J. Math. Anal. Appl.. 375, 99107## B. Zhou and Z. Du (2010). On eccentric connectivity index. MATCH Commun. Math. Comput. Chem.. 63, 181198## ]