A CHARACTRIZATION OF GRAPHS WITH 3-PATH COVERINGS AND THE EVALUATION OF THE MINIMUM 3-COVERING ENERGY OF A STAR GRAPH WITH M RAYS OF LENGTH 2

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1 A CHARACTRIZATION OF GRAPHS WITH -PATH COVERINGS AND THE EVALUATION OF THE MINIMUM -COVERING ENERGY OF A STAR GRAPH WITH M RAYS OF LENGTH PAUL AUGUST WINTER DEPARTMENT OF MATHEMATICS, UNIVERSITY OF KWAZULU NATAL, HOWARD COLLEDGE, GLENWOOD,DURBAN, SOUTH AFRICA, 44 eail winter@ukzn.ac.za ABSTRACT The sallest set Q of vertices of a graph G, such that every path on vertices, has at least one verte in Q, is a iniu -covering of G. By attaching loops of weight to the vertices of G we can fin the eigenvalues associate with G, an hence the iniu -covering energy of G. In this paper we characterize graphs with -coverings in ters of non-q-covere eges, an we eterine the iniu -covering energy of a star graph with rays each of length.. INTRODUCTION The Huckel Molecular Orital theory provie the otivation for the iea of the energy of a graph the su of the asolute values of the eigenvalues associate with the graph see [. This resulte in the iea of the iniu -covering energy of a graph in [ an the iniu -covering energy of star graphs with rays of length were foun. This iea was generalize in [ an the iniu - covering energy of coplete graphs were eterine. In this paper we eterine the iniu -covering energy of star graphs with rays of length.

2 . RESULTS OF MINIMUM COVERINGS OF GRAPHS All graphs which we shall consier will e finite, siple, loopless an unirecte. Let G e such a graph of orer n with verte set v, v,..., v }. A covering - { n covering of a connecte graph G is a set S of vertices of G of such that every ege of G has at least one verte in S. Since an ege is a path length on vertices a -path we generalize this in [, in ters of energy, y introucing a -covering or -path covering of a graph G as eing set Q of vertices of G such that every path of G of length or -path has at least one verte in Q. Any -covering set of G of iniu carinality is calle a iniu -covering of G. In the following theores, Q is a -covering of G, an if the verte of G, which is not in Q, is a penant verte verte of egree of G, that elongs to a path P u,v,w,,y, of length s fro Q, where u is the only verte in Q, we say that P is with respect to Q a s-penant path of G an the ege y is with respect to Q the s-penant ege of P, an y the ile verte of a -penant path of G. If a verte u is in Q, then the istance of u fro Q is taken as. THEOREM No verte of G can e a istance of ore than fro Q. Proof Suppose the verte u of G, that is not in Q, is a istance fro Q. Then there eists a path uvw of length on 4 vertices such that vertices u,v,w are not in Q, ut is in Q. We therefore have a path uvw of length with u,v an w not in Q, which is a contraiction.

3 THEOREM If u is a verte that is a istance fro Q, then u is a penant verte. Proof Suppose u is on the path uvw of length, where vertices u,v are not in Q an w is, an no shorter path eist fro u to Q. If u is not a penant verte, then it ust e connecte to a verte y, where y is not in Q. But then it follows that we have a path vuy on vertices u,v an w which oes not have any verte in Q, a contraiction. THEOREM If uv is an ege of G where u an v o not elong to Q, an neither are penant vertices, then uv is the ile of a path uvw P, of length on 4 vertices, such that the ens an w of the path are oth in Q, an/or uv is the ege of the triangle uv where is in Q. These eges are isjoint or overlap in oth vertices. Proof Let uv e an ege of G, neither of which elong to Q or are penant vertices. Then there ust eist vertices w an y such that wuvy is a path on 4 vertices in G. If w is not in Q, then we have a path wuv on vertices with no verte in Q, a contraiction. Siilarly if y is not in Q we get a path uvw on vertices with no verte in Q. Thus w an y ust elong to Q. If w an y are istinct, then we get the 4-path case, otherwise we get the triangle case. Suppose two such eges uv an u v have eactly verte in coon, say v u. Then we have a path uv u where neither u,v or u are in Q, a contraiction. Thus the eges can overlap in oth vertices or they ust e isjoint.

4 The path on 4 vertices in theore is efine as a -covering hanle-path of G an the non-covere ege uv the ile ege of this hanle-path. The triangle in theore is a -covering triangle of G, an the non-covere ege, uv, the triangle ege of G. The ege uv in each case is a non-covere ege with respect to Q of G 4 THEREM 4 If u is a verte of G then either u is in Q,or if u is not in Q then. u is a penant verte of a -penant path or a -penant path or. if u is not a penant verte, then.. u is the ile verte of a -penant path vuw where v is in Q an w is a penant verte an/or.. u elongs to the path uv on vertices where an v are in Q efine as a -covering V path of G an/or.. u elongs to the ile ege of a -covering hanle-path of G an/or u elongs to the triangle ege of a -covering triangle of G. THEOREM 5 If u is a verte of G that is not in Q an is a penant verte, then u is a penant verte of either a -penant path or a -penant path of G. Thus no s-penant paths eist of length greater than. THEOREM If uv is an ege of G, where neither u nor v elongs to Q i.e. a non-covere ege of G, then either uv is a penant ege of a -penant path of G, or uv is the ile ege of a -covering hanle- path of G, or uv is the triangle ege of a - covering triangle of G; the eges ust e isjoint, or overlap in oth vertices in the case of the non-penant eges. Thus there are only types of non-q-covere eges of a graph G with a - covering set Q- a -penant ege, a ile ege of a hanle path, an a triangle

5 ege referre to as a -penant, hanle or triangle ege. These eges are isjoint ecept for the eges of the non-penant kin which can overlap in oth vertices. If a -penant ege uv has v in coon with a hanle or triangle ege vw, then we will have a path uvw with no verte in Q. THEOREM 7 A graph G has a -covering Q if an only if the non-q-covere eges of G are either -penant, hanle or triangle eges which are isjoint ecept for nonpenant eges which can overlap in oth vertices. 5. MOLECULAR STRUCTURES AND ENERGY The iniu -covering energy of olecular structures given in [ involves the sallest set of atos, such that every ato of the structure, is either in the set, or is connecte via ons irectly to at least one verte of the set. This is generalize to a iniu -covering energy of olecular structures, where the sallest set Q of atos is consiere, such that every ato, is either in the set, or connecte y a path of one atos of length at ost, to at least one ato in the set. If two atos u an v are one y the ege uv, an neither u an v are in Q, then we say the pair of one atos are non-q-covere. In ters of energy of a structure, in orer, say, to prevent estailization, we ay seek the sallest set Q of atos to e energize, such that all non-q-covere one atos are either, as eges, -penant, hanle or triangle eges of the structure where the eges are isjoint an the non-penant eges ay overlap in oth vertices.

6 4. THE MINIMUM--COVERING ENERGY OF A GRAPH A iniu -covering atri of G with a iniu -covering set Q of vertices is a atri A Q G a i, where j if vv i j E G a ij if i j an v Q i * otherwise The ile conition * is equivalent to loops of weight eing attache to the vertices of Q. G The characteristic polynoial of A Q is then enote y f n G, et I A G Q The iniu -covering energy See [ is then efine as E Q G n i Where i the iniu -covering eigenvalues are the n real roots of the characteristic polynoial. 5. THE GENERALIZED STAR GRAPH The star graph on verices with rays of length, can e generalize to a star graph with rays of length n-

7 Take copies of the pathp n, join the paths at their en vertices,in the centre verte u, enote the graph on n n n vertices y S, P ;, n n If n an, then we get the star graph 7 K, with rays of length which has a iniu -covering eigenvalues the sae as the iniu - K, covering eigenvalues of ; the two non-zero eigenvalues are given in [ 4 4 ; K, is 4. which iplies that the iniu -covering energy of. THE STAR GRAPH WITH RAYS OF LENGTH If n an, then we lael the vertices of the star graph of length on vertices as follows S with rays P, Centre verte is u, the set of vertices a istance, respectively, fro u is laele V { v, v,... v },; V { v, v,... v } respectively. The possile -covering sets are { u}; V- ut the iniu -covering is the forer set. For constructing the ajacency atri A of the S we lael the center u as v, P, the vertices of V,V as V { v, v,... v },; V { v, v,..., v } respectively.

8 8 For, n we have the path 5 P with iniu -covering } { v Q with iniu -covering ajacency atri, P Q S A so that the characteristic equation is 5 5, et et P A Q S I an epaning using first row Epaning the last atri eterinants aout the r an 4 th rows Epaning the last two atri eterinants aout the st an n rows respectively

9 The first eterinant involves the circulant atri with solutions πij ep 4 ep πij; n,,,. The secon eterinant involves the circulant atri with solutions πij ep ep πij; n,. Thus the characteristic equation is [ [ [ [ We generalize this to fining the characteristic equation of a star graph on vertices with rays of length For an n we have vertices u, v, v,..., v, v, v,..., v

10 , P S A so that the characteristic equation, et et P S A I Epaning the eterinant using the first row

11 Followe y eterinants... Epaning the last atri eterinants aout the th, th,, th rows respectively

12 .. Last eterinants epaning using row, respectively The first atri coes fro the circulant atri with eigenvalues ties ties j ij ;; ;,,,,.., ; [ep π

13 The secon atri coes fro the circulant atri with eigenvalues πij [ep ; j,,,..,, ; ties;; ties Which yiels the characteristic equation [ [ [ Thus the graph has iniu -covering eigenvalues an -, each of ultiplicity -, an eigenvalues fro the roots of the cuic equation THEOREM 8 The iniu -covering energy of the star graph with rays of length Is where., are the roots of the cuic equation The roots can e foun in cuic functions in Wikipeia [ c 7 c 7 4 c

14 [ c c c [ c c c [ c c c [ c c c [ c c c With a, -, c- an [ [ [ [

15 [ [ Siplifying [ [ [ [ [

16 [ REFERENCES. Aiga C., Baya, A., Gutan, I., Srinivas, S. A. THE MINIMUM COVERING ENERGY OF A GRAPH. Kragujevac J. Sci. 4, Winter, P. A. THE MINIMUM -COVERING ENERGY OF COMPLETE GRAPHS. arxiv v..