Iteratioal Joural of Combiatorial Graph Theory ad Applicatios Vol 4, No 2, (July-December 2011), pp 77-87 ENERGY, DISTANCE ENERGY OF CONNECTED GRAPHS AND SKEW ENERGY OF CONNECTED DIGRAPHS WITH RESPECT TO ITS SPANNING TREES AND ITS CHORDS Murugesa N & Meeakshi K ABSTRACT: The eergy of a graph G is defied as the sum of the absolute values of the eigevalues of its adjacecy matrix The distace eergy of a graph G is defied as the sum of the absolute values of the D-eige values of its distace matrix The skew eergy of a graph G is defied as the sum of the absolute values of the eigevalues of its skew adjacecy matrix Jae Day ad Wasi So, [2007] have discussed about the chages i the eergy of the graph due to the deletio of edges I this paper the chages of eergy, distace eergy of coected graphs ad the skew eergy of coected digraphs are studied with respect to its spaig trees ad its chords AMS CLASSIFICATION: 15A45, 05C50 KEYWORDS: Coected graphs, Coected digraphs, Spaig trees, Graph eergy, Distace eergy, Skew eergy 1 INTRODUCTION Graphs ca be used to represet ay physical situatio ivolvig discrete objects amog them I geeral, graph theory has its impact o each ad every brach of sciece I Mathematics ad Computer Sciece, coectivity is oe of the basic cocepts of graph theory It is closely related to the theory of etwork flow problems The coectivity of a graph is a importat measure of its robustess as a etwork Oe of the importat facts about coectivity i graphs is Meger s theorem, which characterizes the coectivity ad edge coectivity of a graph i terms of the umber of idepedet paths betwee vertices Coected graphs are useful for characterizatio ad idetificatio of chemical compouds A coected graph is used to represet a chemical compoud with vertices as the atoms ad the edges for represetig the bods betwee the atoms Cayley showed that every group of order ca be represeted by a strogly coected digraph of vertices, i which each vertex correspods to a group elemet ad edges carry the label of a geerator of the group Cayley used edges of differet colors to show differet geerators We ca say that a graph of a cyclic group of order is a directed circuit of vertices i which every vertex has the same label Also the digraph of a group uiquely defies the group by specifyig how every product of elemets correspods to a directed
78 MURUGESAN N & MEENAKSHI K edge sequece This digraph kow as the Cayley diagram is useful i visualizig ad studyig abstract groups The cocept of eergy of a graph arose i chemistry, where certai umerical quatities, such as the heat formatio of a hydrocarbo are related to the so-called total π-electro eergy that ca be calculated from the eergy of the correspodig molecular graph The eergy of a simple graph G was first defied by Iva Gutma [12] i 1978 as the sum of the absolute values of the eigevalues of its adjacecy matrix The molecular graph is the represetatio of molecular structure of a hydrocarbo, whose vertices are the positio of carbo atoms ad two vertices are adjacet, if there is a bod coectig them I chemistry, molecular orbital theory (MO) is a method for determiig molecular structure i which electros are ot assiged to idividual bods betwee atoms, but are treated as movig uder the ifluece of the uclei i the whole molecule I this theory, each molecule has a set of molecular orbitals i which it is assumed that the molecular orbital wave fuctio Ψ f may be writte as a simple weighted sum of the costituet atomic orbitals χ i accordig to the followig equatio Ψ = c χ j ij i i = 1, where the co-efficiets c ij may be determied umerically usig Schrodiger equatio ad applicatio of the variatioal priciple The Huckel method or Huckel molecular orbital method (HMO) proposed by Erich Huckel i 1930, is a very simple liear combiatio of atomic orbitals-molecular orbitals (LCAO-MO) method for the determiatio of eergies of molecular orbitals of pi electros i cojugated hydrocarbo systems, such as ethae, bezee ad butadiee The exteded Huckel method developed by Roald Hoffma is the basis of the Woodward-Hoffma rules The eergies of exteded cojugated molecules such as pyridie, pyrrole ad fura that cotai atoms other tha carbo also kow as heteroatoms had also bee determied usig this method Huckel expressed the total π-electro of a cojugated hydrocarbo as π /2 i = 1 E = α + 2β λ where α ad β are costats ad the eigevalues pertai to a special so called molecular graph like ethylee, bezee, butadiee, cyclo-buta-di-ee, etc For the sake of simplicity, the expressio for E π is give, whe is eve The oly o-trivial part i the above formula for E π is /2 2 i It reduces to graph eergy provided λ 0 λ [3] 1λ i = i + 1 2 2 The complete graph K has eigevalues 1 ad 1 ( 1 times) At oe time it was thought that the complete graph K has the largest eergy amog all vertex
ENERGY, DISTANCE ENERGY OF CONNECTED GRAPHS AND SKEW ENERGY OF CONNECTED 79 graphs G, that is, E (G) 2 ( 1) with equality if ad oly if G = K But Godsil i the early 1980s costructed a example of a graph o vertices whose eergy exceeds 2 ( 1) Graphs whose eergy satisfies E (G) > 2 ( 1) are called hypereergetic If E (G) 2 ( 1), G is called o-hypereergetic I this paper we study about the eergy, distace eergy of coected graphs ad skew eergy of coected digraphs with respect to its spaig trees ad its chords 2 IN THIS SECTION, WE DEFINE SOME BASIC DEFINITIONS OF GRAPH THEORY Defiitio 21: A liear graph or simply a graph G (V, E) cosists of a set of objects V = {v 1, v 2, v 3,, v } called vertices, ad aother set, E = {e 1, e 2, e 3,, e } whose elemets are called edges, such that each edge e k is idetified with a uordered pair (v i, v j ) of vertices Defiitio 22: A directed graph or a digraph G i short cosists of a set of vertices V = {v 1, v 2, v 3,, v }, a set of edges E = {e 1, e 2, e 3,, e }, ad a mappig Ψ that maps every edge oto some ordered pair of vertices (v i, v j ) A digraph is also referred to as a orieted graph Defiitio 23: A graph is said to be coected if there is at least oe path betwee every pair of vertices i G Otherwise the graph is said to be discoected A ull graph of more tha oe vertex is discoected A discoected graph cosists of two or more coected graphs Each of these coected graphs is called a compoet The graph give below is a discoected graph with two compoets which are coected subgraphs Defiitio 24: A simple graph i which there exists a edge betwee every pair of vertices is called a complete graph The complete graph with vertices is deoted by K A complete graph is also referred as a uiversal graph or a clique Sice every vertex is joied with every other vertex through oe edge, the degree of every vertex ( 1) is 1 i a complete graph of vertices Also the total umber of edges i G is Defiitio 25: A graph G is said to be a bipartite graph, if there is a partitio of V (G) ito two subsets A ad B such that o two vertices i the same subset are adjacet If G is a bipartite graph with A ad B as defied above, the [A, B] is called a bipartitio of G; ad G is deoted by G [A, B] 2
80 MURUGESAN N & MEENAKSHI K A bipartite graph G [A, B] is called a complete bipartite graph if every vertex i A is adjacet with every vertex i B If ad A = m ad B =, the the graph is deoted by K m, Defiitio 26: A closed walk i which o vertex (except the iitial ad fial vertex) appears more tha oce is called a circuit That is, a circuit is a closed oitersectig walk Defiitio 27: A tree is a coected graph without ay circuits A tree has to be a simple graph that is havig either a self-loop or parallel edges A graph with atleast oe vertex is also a tree A ull tree is a tree without ay vertices The geealogy of a family ca be represeted by meas of a tree A river with its tributaries ad subtributaries ca be represeted by a tree Trees with two ad three vertices are give below Defiitio 28: A tree T is said to be a spaig tree of a coected graph G if T is a subgraph of G ad T cotais all vertices of G A spaig tree is also referred as a skeleto or scaffoldig of G Sice spaig trees are the largest trees with maximum umber of edges i G, it is also called as the maximal tree subgraph or maximal tree of G The spaig trees K 3 of are give below: b b b a c a c a c Defiitio 29: A edge i a spaig tree T is called its brach A edge of G which is ot i the give spaig tree is called a chord Braches ad chords are defied oly with respect to a give spaig tree A edge that is a brach of oe spaig tree i a graph may be a chord with respect to aother spaig tree b a c a c The edges {a, b} ad {b, c} are the braches of the spaig tree of the complete graph K 3 The edge {a, c} which is ot i the spaig tree is a chord
ENERGY, DISTANCE ENERGY OF CONNECTED GRAPHS AND SKEW ENERGY OF CONNECTED 81 Defiitio 210: The uio of two graphs G 1 = (V 1, E 1 ) ad G 2 = (V 2, E 2 ) is aother graph G 3 writte as G 3 = G 1 G 2 whose vertex set V 3 = V 1 V 2 ad the edge set E 3 = E 1 E 2 The uio operatio is commutative That is, G 2 G 1 = G 1 G 2 A coected graph G ca be expressed as the uio of two subgraphs T ad T, that is G = T T, where T is a spaig tree, ad T is the complemet of G i T The graph T is the collectio of chords, is called as the chord set or the tie set or the co-tree of T It is also kow that a coected graph of vertices ad e edges has 1 tree braches ad e + 1 chords, with respect to ay of its spaig trees 3 IN THIS SECTION WE DISCUSS ABOUT THE BASIC DEFINITIONS OF ENERGY, DISTANCE ENERGY OF CONNECTED GRAPHS AND ALSO THE SKEW ENERGY OF CONNECTED DIGRAPHS Defiitio 31: Let G be a simple, fiite, udirected graph with vertices ad m edges Let A = (a ij ) be the adjacecy matrix of graph G The eigevalues of λ 1, λ 2,, λ of A, assumed i o-icreasig order, are the eigevalues of the graph G The eergy of a graph G, deoted by E (G) is defied as E () G = λ i = 1 i The set {λ 1, λ 2,, λ } is the spectrum of G ad is deoted by G If the distict eigevalues of G are µ 1 > µ 2 > > µ ad if their multiplicities are m (µ 1 ), m (µ 2 ),, m (µ s ) the we write µ 1 µ 2 µ s Spec G = m()() µ 1 m () µ 2 m µ s Spec G is idepedet of labelig of the vertices of G Asis a real symmetric matrix with zero trace, these eigevalues are real with sum equal to zero Defiitio 32: Let G be a simple graph with vertices ad m edges Let G be also coected ad let its vertices be labeled as v 1, v 2,, v The distace matrix of a graph G is defied as a square matrix D = D (G) = (d ij ) where d ij is the distace betwee the vertices v i ad v j i G The eige values of the distace matrix D (G) are deoted by µ 1, µ 2,, µ ad are said to be the D-eigevalues of G Sice the distace matrix is symmetric, its eigevalues are real ad ca be ordered as µ 1 µ 2 µ ad are said to form the D-spectrum of G The distace eergy or the D-eergy of a graph G, deoted by E D or E D (G) is defied as E = µ ad it was first itroduced by Idulal et al D i = 1 i
82 MURUGESAN N & MEENAKSHI K Defiitio 33: Let D be a simple digraph of order with the vertex set V (D) = {v 1, v 2,, v } ad arc set Γ (D) V (D) V (D) We have (v i, v i ) Γ (D) for all i, ad (v i, v j ) Γ (D) implies that (v j, v j ) Γ (D) The skew-adjacecy matrix of the digraph D is the matrix S (D) = (s ij ) where s ij = 1, wheever (v i, v j ) Γ (D), s ij = 1, wheever ( v j, v i ) Γ (D), ad s ij = 0 otherwise S (D) is also the skew-symmetric matrix The eigevalues {λ 1, λ 2,, λ } of S (D) are all purely imagiary umbers ad the sigular values of S(D) coicide with the absolute values { λ 1, λ 2,, λ } of its eigevalues Cosequetly the eergy of S (D), which is defied as the sum of its sigular values [1], is also the sum of the absolute values of its eigevalues This eergy is called as the skew eergy of the digraph D, deoted by E s (D) Therefore the skew-eergy of a digraph is defied as E () D = λ Theorem 34: The eergy of coected graph G decreases with respect to the sum of the eergies of its spaig tree ad the chord That is E ()()() G < E T + E T Proof: Let G be ay coected graph with vertices ad edges Let T ad T be its spaig tree ad chord Jae Day ad Wasi So have show that with deletio of a edge i a coected graph G, the eergy of coected graph G icreases, decreases or remais the same s i = 1 A coected graph G of vertices has 1 braches ad e + 1 chords The spaig tree T has 1 edges ad the chord T has e + 1 edges Sice the umber of edges of the spaig tree T ad the chord T is less tha that of the coected graph G, the eergy of the spaig tree T is either less tha or greater tha or equal to that of the coected graph G It is see that i all the coected graphs G, the eergy of the chord T is less tha that of the eergy of the coected graph G But the sum of the eergies of the spaig tree T ad the chord T of the coected graph G is greater tha that of the eergy of the coected graph G except for the star graph That is the eergy of the coected graph G decreases with respect to the sum of the eergies of ay of its spaig tree ad the chord We show that the above statemet is true for differet types of coected graphs i
ENERGY, DISTANCE ENERGY OF CONNECTED GRAPHS AND SKEW ENERGY OF CONNECTED 83 Case 1: Let be the complete graph K We kow that a coected graph G of vertices ad e edges has 1 tree braches ad e + 1 chords with respect to ay of its spaig tree Jae Day ad WasiSo have show that the deletio of a edge i a complete graph K decreases its eergy That is E (K ) > E (K {e}) We kow that the umber of spaig trees of the complete graph is 2 The chord T of the complete graph K is a discoected graph with some ull vertices Sice the umber of edges of the spaig tree ad chord of the complete graph K is less tha, the eergies of the spaig tree ad the chord of the complete graph K is less tha that of the eergy of the complete graph K Case 2: Let G be the cycle C of legth K We kow that a cycle is possible from a coected graph with vertices ad m edges if ad oly if m The eergy of ay spaig tree of the cycle C is either less tha or greater tha that of the eergy of the cycle C The complemet T of the spaig tree T of C is a complete graph K 2, whose eergy is 2 That is E (T ) = 2 Case 3: (i) Let G be oly the bipartite graph K m, If G is a bipartite graph with > 2 vertices ad m 2 edges, the the eergy 2m of the bipartite graph is defied as ( ) + 2m E() G 2( 2) 2 2 m ( ) ad the equality case is characterized i some special cases [15] The eergy of a graph as a fuctio of its m umber of edges satisfies 2() m 2 E G m where the equality o the left holds if ad oly G is a complete bipartite graph ad o the right holds if ad oly if G is a matchig of m edges [3] The eergy of ay spaig tree T of the bipartite graph K m, is see to be less tha that of the eergy of the bipartite graph K m, The complemet of the spaig tree T of the bipartite graph K m, is always a discoected graph This discoected graph is the uio of the complete graphs K 2 ad some isolated vertices The eergy of the chord T is sum of 2
84 MURUGESAN N & MEENAKSHI K the eergies of the discoected compoets Sice the eergy of the complete graph K 2 is a multiple of 2, the eergy of the chord T is also a multiple of 2 (ii) Let G be the complete bipartite graph K m, We kow that for the complete bipartite graph, E () Km, 2 = is the umber of edges m where m The eergy of ay spaig tree T of the complete bipartite graph K m, is less tha that of the eergy of the complete bipartite graph K m, The complemet of the spaig tree T of the complete bipartite graph K m, is a discoected graph with some isolated vertices The eergy of is the sum of the eergies of T its discoected compoets (iii) Let G be the complete regular bipartite graph K, We kow that the eergy of the complete regular bipartite graph K m, is 2 The eergy of a complete regular bipartite graph icreases with deletio of a edge from it That is E (K, ) < E (K, {e}) The eergy of the spaig tree T of the regular complete bipartite graph K, is greater tha that of the eergy of the complete bipartite graph K, The complemet of the spaig tree T of the regular complete bipartite graph K, is a discoected graph The eergy of T is the sum of the eergies of its discoected compoets Case 4: Cosider the star graphs Cosider the star graphs K 1, 1 We kow that a star graph ad its edge deleted graph are complete bipartite graphs The star graph K 1, 1 uiquely has the smallest eergy amog all graphs with vertices where oe of the vertices are isolated [3] The eergy of the star graph is defied as E () K 2 = 1 [3] Every star graph is itself a spaig tree 1, 1 The eergies of the star graph K 1, 1 ad its spaig tree T are same That is E (K 1, 1 ) = E (T) Also the chord of the star graph K 1, 1 is a ull graph
ENERGY, DISTANCE ENERGY OF CONNECTED GRAPHS AND SKEW ENERGY OF CONNECTED 85 This implies that the eergy of the co-tree is zero That is E (T ) = 0 Therefore we have E (K 1, 1 ) = E (T) + E (T ) or E (K 1, 1 ) = E (T) I all the cases, we fid that the eergy of the coected graph G to be less tha or equal to the sum of the eergies of the spaig tree T ad the chord T of the coected graph G That is E (G) < E (T) + E (T ) Theorem 35: The distace eergy of a coected graph is always less tha that of its spaig tree That is E D (G) < E D (T) Proof: Let G be the coected graph ad T be its spaig tree The distace eergy of a coected graph G is greater tha or equal to where is the umber of vertices [18] ( 1), The distace eergy of the complete graph is same as that of the eergy of the complete graph K That is E D (K ) < E (K ) The distace eergy of the complete bipartite graph K m, is give by E D (K m, ) = 4 (m + 2) for m, 2 [19] The distace eergy of a coected graph G always less tha or equal to its edge deleted graph Sice the umber of edges i a spaig tree T of ay coected graph is 1, the distace eergy of the spaig tree T of ay coected graph G will be greater tha the distace eergy of the coected graph The distace eergy of the chord T of coected graphs like complete graphs, bipartite graphs caot be foud as the chord graphs are all discoected graphs I case of star graphs, the chord graph is a ull graph Theorem 36: The skew eergy of coected digraph G decreases with respect to the sum of the skew eergies of its spaig trees ad the chords That is E s (G) < E s (T) + E s (T ) Proof: Let G be the coected digraph Let T be the spaig tree of the digraph G, ad let T be the chord of the coected digraph G
86 MURUGESAN N & MEENAKSHI K The skew eergy of the directed tree is same as that of the eergy of the udirected tree [1] We ca say that the skew eergies of the spaig directed tree T ad the eergy of the spaig udirected tree T are same The complemet T of the directed tree T of the coected digraph G is a discoected digraph with some isolated vertices The skew eergy of the discoected digraph is the sum of the skew eergies of its discoected compoets The skew eergy of the coected digraph G is see to be less tha that of the sum of the skew eergies of the spaig tree T ad the co-tree T Therefore we ca say that E S (G) < E S (T) + E S (T ) We ca verify the above two theorems with the examples of some coected graphs ad coected digraphs with respect to oe of its spaig trees ad chord give i tabular form Coceted graph G E (G) E (T) E (T ) E D (G) E D (T) E S (G) E S (T) E S (T ) K 3 4000 2828 2000 4000 5464 3464 2828 2000 K 4 6000 4472 4472 6000 10325 5656 4472 4472 C 3 4000 2828 2000 4000 5464 3464 2828 2000 C 4 4000 4472 2000 8000 10324 5656 4472 2000 C 5 4472 5464 2000 12000 16576 6156 5464 2000 K 2, 3 4898 5226 2828 10671 14919 6828 5226 2828 K 2, 3 5596 5226 2000 13076 14919 6293 5226 2000 K 3, 4 8120 7880 4000 27401 29728 8120 7880 4000 K 2, 2 4000 4472 2000 8000 10325 4000 4472 2000 K 1, 4 4000 4000 0000 13212 13212 4000 4000 0000 ACKNOWLEDGEMENT I thak my guide ad Karpagam Uiversity for givig me a opportuity to carry o with my research work REFERENCES [1] Adiga C, Balakrisha R, ad Wasi So, The Skew Eergy of a Digraph, Liear Algebra ad its Applicatios, (2009) [2] Balakrisha R, Liear Algebra ad Its Applicatios, 387, (2004), 287-295 [3] Richard Brualdi, Eergy of a Graph, A Discussio of Graph Eergy for the AIM Workshop o Spectra of Families of Matrices Described by Graphs, Digraphs ad Sig Patters, (October 23-27, 2006)
ENERGY, DISTANCE ENERGY OF CONNECTED GRAPHS AND SKEW ENERGY OF CONNECTED 87 [4] Edelberg M, Garey M R, ad Graham R L, O the Distace Matrix of a Tree, Discrete Mathe, 14, (1976), 23-39 [5] Graham R L, ad L Lovasz, Distace Matrix Polyomial of Trees, Adv Math, 29, (1978), 60-88 [6] Graham R L, ad H O Pollak, O the Addressig Problem for Loop Switchig, Bell System Tech J, 50, (1971), 2495-2519 [7] Harrary, Graph Theory [8] Hoffma A J, McAdrew M H, The Polyomial of a Directed Graph, Proc Amer Math Soc, 16, (1965), 303-309 [9] Hosoya H Murakami, ad Gotoh M, Distace Polyomial ad Characterizatio of a Graph, Natur Sci Rept Ochaumizu Uiv, 24, (1973), 27-34 [10] Hua H, O Miimal Eergy of Uicyclic Graphs with Prescribed Girth ad Pedat Vertices, MATCH Commu Math Comput Chem, 57, (2007), 351-361 [11] Idulal G, Gutma I, ad Vijaykumar, O the Distace Eergy of a Graph, MATCH Commu Math Comput Chem, 60, (2008), 461-472 [12] Iva Gutma, Graph Eergy, Laplacia Graph Eergy ad Beyod [13] Jae Day, ad Wasi So, Graph Eergy Chage Due to the Edge Deletio, A Discussio of Graph Eergy for the AIM Workshop o Spectra of Families of Matrices Described by Graphs, Digraphs ad Sig Patters, (October 23-27, 2006) [14] Jae Day, ad Wasi So, Sigular Value Iequality ad Graph Eergy Chage, Electroic Joural of Liear Algebra, 16, (September 2007), 291-299 [15] Koole H Jack, ad Vicet Moulto, Maximal Eergy Bipartite Graphs, Graphs ad Combiatorics, (2003), pp 19 & 131-135 16] Murugesa N, ad Meeakshi K, Sigular Value Iequality ad Relatio betwee Eergy Distace Eergy ICSE Coferece, IEEE Proceedigs, April 2010), p 272 [17] Narsigh Deo, Graph Theory with Applicatios to Egieerig ad Computer Sciece [18] Ramae H S, Revakar D S, Iva Gutma, Siddai Bhaskara Rao, Acharya B D, ad Walikar H B, Bouds for the Distace Eergy of a Graph, Kragujevac J Math, 31, (2008), 59-68 [19] Stevaovic D, ad Idulal G, Applied Mathematics Letters, 22, (2009), 1136-1140 [20] Walikar H B, Gutma I, Revakar D S, ad Ramae H S, Distace Spectra ad Distace Eergies of Iterated Lie Graphs of Regular Graphs, Publicatios Del Istitut Mathemaque, Nouvelle Serie, Tome, 85/99, (2009), 39-46 Murugesa N Govermet Arts College, Coimbatore, Idia Meeakshi K CMR Istitute of Techology, Bagalore, Idia E-mail: krishapriya531@yahoocom