AUGUST WINTER, PAUL

Research Publication Journals Repository

ADV  MATH
SCI  JOURNAL
Advances in Mathematics: Scientic Journal 3 (2014), no.2, 97107
ISSN 1857-8365
A CHARACTERIZATION OF GRAPHS WITH 3-PATH COVERINGS
AND THE EVALUATION OF THE MINIMUM 3-COVERING ENERGY
OF COMPLETE GRAPHS AND STAR GRAPH WITH M RAYS OF
LENGTH 2
PAUL AUGUST WINTER
Abstract. The smallest set Q of vertices of a graph G, such that every path on 3
vertices has at least one vertex in Q, is a minimum 3-covering of G. By attaching
loops of weight 1 to the vertices of Q, we can nd the eigenvalues associated with G,
and hence the minimum 3-covering energy of G. In this paper we characterize graphs
with 3-coverings in terms of non Q-covered edges, and we determine the minimum
3-covering energy of complete graphs and star graph with m rays each of length 2.
1. Introduction
1.1. Energy. The Huckel Molecular Orbital theory provided the motivation for the idea
of the energy of a graph- the sum of the absolute values of the eigenvalues associated
with the graph (see [1]). This resulted in the idea of the minimum 2-covering energy of
a graph in [1] and the minimum 2-covering energy of star graphs with m rays of length
1 were found. This idea is generalized in this paper where we introduce the concept
of a minimum 3-covering of a graph. We characterize such graphs and determine the
minimum 3-covering energy of complete graphs and star graphs with m rays of length 2.
1.2. Motivation for 2 and 3-covering energy of a graph. If one considers a molecule
in a graph-theoretical way, where the atoms are the vertices and the edges the bonds
between the atoms (see [2])), then the idea of energizing the whole molecule is relevant.
Conserving energy will involve the smallest set Q of atoms which can be energized so that
all atoms outside Q, and connected to at least one element of Q will also be energized.
This is equivalent, graphically, to nding a minimum 2-covering of a graph, i.e. every
path of length 1 has at least one end in Q. Thus, in a similar way, the smallest set Q
of atoms that can aect vertices, not in Q, on paths of length 2 will imply a minimum
3-covering.
2010 Mathematics Subject Classication. 05C50.
Key words and phrases. Vertex coverings of graphs, minimum vertex coverings, energy of graphs,
eigenvalues and coverings of graphs.
97
98 PAUL AUGUST WINTER
2. Results of minimum 3-coverings of graphs
All graphs which we shall consider will be nite, simple, and loopless and undirected
and graph theoretical notation of [3] will be adopted. Let G be such a graph of order
n with vertex set fv1; v2; : : : ; vng. A covering (2-covering) of a connected graph G is a
set S of vertices of G such that every edge of G has at least one vertex in S. Since an
edge is a path length 1 on 2 vertices (a 2-path), we generalize this in terms of energy, by
introducing a 3-covering (or 3-path covering) of a graph G.
Denition 2.1. A 3-covering (or 3-path covering) of a graph G is a set Q of vertices
of G such that every path of G of length 2 (or 3-path) has at least one vertex in Q.
Any 3-covering set of G of minimum cardinality is called a minimum 3-covering of
G.
In the following theorems, Q is a 3-covering of G, and if the vertex x of G, which
is not in Q, is a pendant vertex (vertex of degree 1) of G, that belongs to a path P =
fu; v;w; : : : ; y; xg of length s from Q, where u is the only vertex in Q, we say that P is
(with respect to Q) a s-pendant path of G and the edge yx is (with respect to Q) the
s-pendant edge of P , and y the middle vertex of a 3-pendant path of G. If a vertex u is
in Q, then the distance of u from Q is taken as 0.
Lemma 2.1. No vertex of G can be a distance of more than 2 from Q.
Proof. Suppose the vertex u of G, that is not in Q, is a distance 3 from Q. Then there
exists a path uvwx of length 3 on 4 vertices such that vertices u; v; w are not in Q, but
x is in Q. We therefore have a path uvw of length 3 with u; v and w not in Q, which is
a contradiction. 
Lemma 2.2. If u is a vertex that is a distance 2 from Q, then u is a pendant vertex.
Proof. Suppose u is on the path uvw of length 2, where vertices u; v are not in Q and
w is, and no shorter path exist from u to Q. If u is not a pendant vertex, then it must
be connected to a vertex y, where y is not in Q. But then it follows that we have a path
vuy on 3 vertices u; v and w which does not have any vertex in Q, a contradiction. 
Lemma 2.3. If uv is an edge of G where u and v do not belong to Q, and neither are
pendant vertices, then uv is the middle of a path xuvw P , of length 3 on 4 vertices,
such that the ends x and w of the path are both in Q, and/or uv is the edge of the
triangle xuv where x is in Q. These edges are disjoint or overlap in both vertices
Proof. Let uv be and edge of G, neither of which belong to Q or are pendant vertices.
Then there must exist vertices w and 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 3 vertices with no vertex in Q, a contradiction.
Similarly if y is not in Q we get a path uvw on 3 vertices with no vertex in Q. Thus w and
y must belong to Q. If w and y are distinct, then we get the 4-path case, otherwise we get
the triangle case. Suppose two such edges uv and uv have exactly 1 vertex in common,
say v = u. Then we have a path uvu where neither u; v or u are in Q, a contradiction.
Thus the edges can overlap in both vertices or they must be disjoint.
The path on 4 vertices in theorem 3 is dened as a 3-covering handle-path of G and
the (non-covered )edge uv the middle edge of this handle-path. The triangle in lemma
A CHARACTERIZATION OF GRAPHS... 99
(2.3), is a 3-covering triangle of G the (non-covered) edge, uv, the triangle edge of G.
The edge uv in each case is a non-covered edge (with respect to Q) of G. 
Lemma 2.4. If u is a vertex of G then either:
(1) u is in Q, or
(2) if u is not in Q then:
 u is a pendant vertex of a 2-pendant path or a 1-pendant path or
 if u is not a pendant vertex then either (i) u is the middle vertex of a 2-
pendant path vuw where v is in Q and w is a pendant vertex and/or (ii)
u belongs to the path xuv on 3 vertices where x and v are in Q (dened
as a 3-covering V path of G) and/or (iii) u belongs to the middle edge
of a 3-covering handle-path of G and/or (iv) u belongs to the triangle
edge of a 3-covering triangle of G.
Corollary 2.1. If u is a vertex of G that is not in Q and is a pendant vertex, then
u is a pendant vertex of either a 2-pendant path or a 1-pendant path of G. Thus no
s-pendant paths exist of length greater than 2.
Lemma 2.5. If uv is an edge of G, where neither u nor v belongs to Q (i.e. a
non-covered edge of G, then either uv is a pendant edge of a 2-pendant path of G,
or uv is the middle edge of a 3-covering handle- path of G, or uv is the triangle edge
of a 3-covering triangle of G; the edges must be disjoint, or overlap in both vertices
in the case of the non-pendant edges.
Thus there are only 3 types of non-Q-covered edges of a graph G with a 3-covering set
Q- a 2-pendant edge, a middle edge of a handle path, and a triangle edge referred to as a
2-pendant, handle or triangle edge. These edges are disjoint except for the edges of the
non-pendant kind which can overlap in both vertices. (If a 2-pendant edge uv has v in
common with a handle or triangle edge vw, then we will have a path uvw with no vertex
in Q.)
Theorem 2.1. A graph G has a 3-covering Q if and only if the non-Q-covered edges
of G are either 2-pendant, handle or triangle edges which are disjoint except for
non-pendant edges which can overlap in both vertices.
3. Molecular structures and energy
The minimum 2-covering energy of molecular structures given in [1] involves the small-
est set of atoms, such that every atom of the structure, is either in the set, or is connected
(via bonds) directly to at least one vertex of the set. This is generalized to a minimum 3-
covering energy of molecular structures, where the smallest set Q of atoms is considered,
such that every atom, is either in the set, or connected by a path (of bonded atoms) of
length at most 2, to at least one atom in the set.
If two atoms u and v are bonded by the edge uv, and neither u and v are in Q, then
we say the pair of bonded atoms are non-Q-covered. In terms of energy of a structure,
in order, say, to prevent destabilization, we may seek the smallest set Q of atoms to
be energized, such that all non-Q-covered bonded atoms are either, as edges, 2-pendant,
100 PAUL AUGUST WINTER
handle or triangle edges of the structure (where the edges are disjoint and the non-pendant
edges may overlap in both vertices).
4. The minimum 3-covering energy of a graph
Denition 4.1. A minimum 3-covering matrix of G with a minimum 3-covering set
Q of vertices is a matrix:
A3Q(G) = (aij)
where:
(4.1) aij =
8<
:
1 if vivj 2 E(G)
1 if i = j& vi 2 Q
0 otherwise
The middle condition (4.1) is equivalent to loops of weight 1 being attached to the
vertices of Q.
The characteristic polynomial is then denoted by:
(4.2) fn(G;) = det(I   A3Q(G))
The minimum 3-covering energy is then dened as:
(4.3) EQ(G) =
nX
1
jj
where i (the minimum 3-covering eigenvalues) are not the n real roots of the character-
istic polynomial.
5. The minimum 3-covering energy of the complete graph
Generally , the minimum 2-covering of a complete graph G on n vertices is any set
of (n   1) vertices of G (see [1]). The minimum 3-covering of a complete graph G on n
vertices is any set of (n  2) vertices. Thus:
(5.1) A3Q(Kn) =
2
666664
1 1 1 : : : 1 1
1 1 1 : : : 1 1
...
...
... 1 1
1 1 1 : : : 0 1
1 1 1 : : : 1 0
3
777775
A CHARACTERIZATION OF GRAPHS... 101
And having the characteristic equation:
(5.2) det(I   A3Q(Kn)) = det
2
666664
  1  1  1 : : :  1  1
 1   1  1 : : :  1  1
...
...
...  1  1
 1  1  1 : : :   1
 1  1  1 : : :  1 
3
777775
Last row minus second last row yields:
= det
2
666664
  1  1  1 : : :  1  1
 1   1  1 : : :  1  1
...
...
...  1  1
 1  1  1 : : :   1
0 0 0 : : :  1   + 1
3
777775
nn
By expanding the last row:
=  (1 + )

1    1  1 : : :  1  1
 1   1  1 : : :  1  1
...
...
...  1  1
 1  1  1 : : :   1  1
 1  1  1 : : :  1  1

=  (+ 1)

  1  1  1 : : :  1  1
 1   1  1 : : :  1  1
...
...
...  1  1
 1  1  1 : : :   1  1
 1  1  1 : : :  1 

= (1 + )n 2   (+ 1)

  1  1  1 : : :  1  1
 1   1  1 : : :  1  1
...
...
...  1  1
 1  1  1 : : :   1  1
0 0 0 : : :   + 1

n 1n 1
= (1 + )n 2   (+ 1)

  1  1  1 : : :  1  1
 1   1  1 : : :  1  1
...
...
...  1  1
 1  1  1 : : :   1  1
 1  1  1 : : :  1  1

n 2n 2
=  (1 + )(+ 1)

  1  1  1 : : :  1  1
 1   1  1 : : :  1  1
...
...
...  1  1
 1  1  1 : : :   1  1
 1  1  1 : : :  1   1

n 2n 2
102 PAUL AUGUST WINTER
(5.3)
=) det(I   A3Q(Kn)) =(1 + )n 2 + (1 + )n 2   (1 + )2n 3(  (n  2))
2(1 + )n 2   (1 + )2n 2 + n 3(1 + )2(n  2)
= (1 + )n 3(2  (1 + )+ (1 + )(n  2))
= (1 + )n 3( 2 + (n  1)+ (n  2))
Eigenvalues are 0 of multiplicity (n  3), -1, and the pair:
(n  1)p(n  1)2 + (4n  8)
 2 =
(n  1)pn2 + 2n  7
2
Theorem 5.1. The minimum 3-covering energy of a complete graph on n  3 ver-
tices is (1 +
p
n2 + 2n  7).
6. The minimum 3-covering energy of the generalized star graph
The star graph on m+ 1 verices with m rays of length 1, can be generalized to a star
graph with m rays of length n  1.
Take m copies of the path, Pn, join the paths at their end vertices, in the centre vertex
u : denote the graph on n+ (n  1)(m  1) = mn m+ 1 vertices by:
S1;mPn ;m  2; n  2 :
If n = 2 and m  3, then we get the star graph, S1;m, with m rays of length 1 which
has a minimum 3-covering eigenvalues the same as the minimum 2-covering eigenvalues
of K1;m; the two non-zero eigenvalues are given in [1]:
1 +
p
4m+ 1
2
;
1 p4m+ 1
2
;
which implies that the minimum 3-covering energy of K1;m is
p
4m+ 1.
7. The star graph with m rays of length 2
If n = 2 and m  2, then we label the vertices of the star graph S1;mP3 with m rays
of length 2 on 2m+ 1 as follows:
Centre vertex is u, the set of m vertices a distance 1; 2 respectively, from u is labeled:
V1 = fv11 ; v12 ; : : : ; v1mg; = V2 = fv22 ; v22 ; : : : ; v2mg ;
respectively.
The possible 3-covering sets are fug;V1 -but the minimum 3-covering is the set fug.
For constructing the adjacency matrix A of the S1;mP3 , we label the center u as v1, the
vertices of V1; V2 as V1 = fv2; v3; : : : ; vm+1g, V2 = fvm+2; vm+3; : : : ; v2m+1g respectively.
A CHARACTERIZATION OF GRAPHS... 103
For n = 3, m = 2, we have the path P5 with minimum 3-covering Q = fv1g with
minimum 3-covering adjacency matrix (where 1 is inserted in the 1; 1 position to represent
the minimum 3-covering):
A3Q(S1;2P3) =
2
66664
1 1 1 0 0
1 0 0 1 0
1 0 0 0 1
0 1 0 0 0
0 0 1 0 0
3
77775 ;
So that the characteristic equation is:
det (I   A3Q(S1;2P3)) = det
2
66664
  1  1  1 0 0
 1  0  1 0
 1 0  0  1
0  1 0  0
0 0  1 0 
3
77775
55
:
By expanding the rst row, we have:
(  1)

 0  1 0
0  0  1
 1 0  0
0  1 0 

44
+

 1 0  1 0
 1  0  1
0 0  0
0  1 0 

44
 

 1   1 0
 1 0 0  1
0  1  0
0 0 0 

44
:
Expanding the last 2 matrix determinants about the 3rd and 4th rows:
(  1)

 0  1 0
0  0  1
 1 0  0
0  1 0 

44
+ 

 1 0 0
 1   1
0  1 

33
  

 1   1
 1 0 0
0  1 

33
:
Expanding the last two matrix determinants about the 1st and 2nd rows respectively:
(  1)

 0  1 0
0  0  1
 1 0  0
0  1 0 

44
  
   1 1 
  
   1 1 
 :
The rst determinant involves the circulant matrix with solutions:
exp(
2ij
4
)2 = exp(ij); j = 0; 1; 2; 3:
The second determinant involves the circulant matrix solutions:
exp(
2ij
2
)1 = exp(ij); j = 0; 1:
104 PAUL AUGUST WINTER
Thus, the characteristic equation is:
(7.1)
(  1)3(+ 2)2   2(  1)(+ 1) = (  1)(+ 1)[(  1)2(+ 1)  2]
= (  1)(+ 1)[(2   1)(  1)  2]
= (  1)(+ 1)[(3   2   + 1)  2]
= (  1)(+ 1)[3   2   3+ 1]
We generalize this to nding the characteristic equation of a star graph on 2m+1 vertices
with m rays of length 2:
For m  2 and n = 3, we have 2m+ 1 vertices: u; v11 ; v12 ; : : : ; v1m; v21 ; v22 ; : : : ; v2m.
A(S1;mP3) =
2
666666664
1 1 1 : : : 0 0
1 0 0 : : : 1 0
1 0 0 : : : 0 1
...
...
... : : :
...
...
0 1 0 : : : 0 0
0 0 1 : : : 0 0
3
777777775
(2m+1)(2m+1)
;
so that the characteristic equation gives:
det (I   A(S1;mP3)) = det
2
666664
  1  1 : : :  1 0
 1  0 : : :  1
 1 0  ... 0
0  1 0  0
0 : : :  1 0 
3
777775
(2m+1)(2m+1)
:
Expanding the determinant using the rst row:
=

  1 : : :  1 0
 1  0 : : :  1
 1 0  ... 0
0  1 0  0
0 : : :  1 0 

(2m)(2m)
;
followed by m determinants:
+

 1 0 : : :  1 0
 1  : : :  1 0
 1 0  ... 0
0  1 0  0
0 : : :  1 0 

 

 1 0 : : :  1 0
 1  : : :  1 0
 1 0  ... 0
0  1 0  0
0 : : :  1 0 

+ : : :+

 1 0 : : :  1 0
 1  : : :  1 0
 1 0  ... 0
0  1 0  0
0 : : :  1 0 

(2m)(2m)
:
A CHARACTERIZATION OF GRAPHS... 105
Expanding the last m matrix determinants about (m + 1)th; (m + 2)th; : : : 2mth rows
respectively:
(  1)

 0 0  1 0 0
0  0 0  1 0
0 0  0 0  1
 1 0 0  0 0
0  1 0 0  0
0 0  1 0 0 

+ 

 1 0 0 0 0
 1  0  1 0
 1 0  0  1
0  1 0  0
0 0  1 0 

(2m 1)(2m 1)
 

 1  0  1 0
 1 0 0 0 0
 1 0  0  1
0  1 0  0
0 0  1 0 

+ : : : + 

 1  0  1 0
 1 0 0 0  1
 1 0  0 0
0  1 0  0
0 0  1 0 

:
Last m determinants expanding using row 1; 2; : : : ;m respectively:
( 1)

 0 0  1 0 0
0  0 0  1 0
0 0  0 0  1
 1 0 0  0 0
0  1 0 0  0
0 0  1 0 0 

(2m)(2m)
 m

 0  1 0
0  0  1
 1 0  0
0  1 0 

(2m 1)(2m 1)
:
The rst matrix comes from the circulant matrix with eigenvalues:
exp[(
2ij
2m
)]m; j = 0; 1; 2; : : : ; 2m  1 = ( 1); ; m times and(1);m times:
The second matrix comes from the circulant matrix with eigenvalues:
exp[(
2ij
2m
)]m 1; j = 0; 1; 2; : : : ; 2m  2 = ( 1);m  1 times and(1);m  1 times ;
which yields the characteristic equation:
(7.2)
(  1)m+1(+ 1)m  m(  1)m 1(+ 1)m 1
= (  1)m 1(+ 1)m 1[(  1)2(+ 1) m]
= (  1)m 1(+ 1)m 1[(2   1)(  1) m]
= (  1)m 1(+ 1)m 1[(3   2   (m+ 1)+ 1]:
Thus, the graph has minimum 3-covering eigenvalues 1 and  1, each of multiplicitym 1,
3 eigenvalues form the roots of the cubic equation:
(7.3) 3   2   (m+ 1)+ 1 = 0:
Theorem 7.1. The minimum 3-covering energy of the star graph with m rays of
length 2 is:
2m  2 + jx1j+ jx2j+ jx3j
where x1; x2; x3 are the roots of the cubic equation:
3   2   (m+ 1)+ 1 = 0:
106 PAUL AUGUST WINTER
The roots can be found by using general method involving cubic:
x1 =   b
3
 1
3
3
r
1
2
[2b3   9bc+ 27d+
p
(2b3   9bc+ 27d)2   4(b2   3c)3]
 1
3
3
r
1
2
[2b3   9bc+ 27d 
p
(2b3   9bc+ 27d)2   4(b2   3c)3]
x2 =   b
3
+
1 +
p
3
6
3
r
1
2
[2b3   9bc+ 27d+
p
(2b3   9bc+ 27d)2   4(b2   3c)3]
+
1 p3
6
3
r
1
2
[2b3   9bc+ 27d 
p
(2b3   9bc+ 27d)2   4(b2   3c)3]
x3 =   b
3
+
1 p3
6
3
r
1
2
[2b3   9bc+ 27d+
p
(2b3   9bc+ 27d)2   4(b2   3c)3]
+
1 +
p
3
6
3
r
1
2
[2b3   9bc+ 27d 
p
(2b3   9bc+ 27d)2   4(b2   3c)3]:
With a = 1, b =  1, c =  (m+ 1), and d = 1:
x1 =
1
3
 1
3
3
r
1
2
[ 2  9(m+ 1) + 27 +
p
( 2  9(m+ 1) + 27)2   4(3m+ 4)3]
 1
3
3
r
1
2
[ 2  9(m+ 1) + 27 
p
( 2  9(m+ 1) + 27)2   4(3m+ 4)3]
x2 =
1
3
+
1 +
p
3
6
3
r
1
2
[ 2  9(m+ 1) + 27 +
p
( 2  9(m+ 1) + 27)2   4(3m+ 4)3]
+
1 p3
6
3
r
1
2
[ 2  9(m+ 1) + 27 
p
( 2  9(m+ 1) + 27)2   4(3m+ 4)3]
x3 =
1
3
+
1 p3
6
3
r
1
2
[ 2  9(m+ 1) + 27 +
p
( 2  9(m+ 1) + 27)2   4(3m+ 4)3]
+
1 +
p
3
6
3
r
1
2
[ 2  9(m+ 1) + 27 
p
( 2  9(m+ 1) + 27)2   4(3m+ 4)3]:
Simplifying:
A CHARACTERIZATION OF GRAPHS... 107
x1 =
1
3
 1
3
3
r
1
2
[16  9m+
p
27m3 + 189m2   144m+ 308]
 1
3
3
r
1
2
[16  9m 
p
27m3 + 189m2   144m+ 308]
x2 =
1
3
+
1 +
p
3
6
3
r
1
2
[16  9m+
p
27m3 + 189m2   144m+ 308]
+
1 +
p
3
6
3
r
1
2
[16  9m 
p
27m3 + 189m2   144m+ 308]
x3 =
1
3
+
1 p3
6
3
r
1
2
[16  9m+
p
27m3 + 189m2   144m+ 308]
+
1 +
p
3
6
3
r
1
2
[16  9m 
p
27m3 + 189m2   144m+ 308]:
References
[1] C. Adiga, A. Bayad, I. Gutman, S. A. Srinivas: The minimum Covering Energy of a Graph,
Kragujevac J. Sci., 34 (2012), 3956.
[2] I. Gutman: The Energy of a Graph: Old and New Results, A. Betten, A. Kohnert, R. Laue, A.
Wassermann (Eds.), Alg. Comb. Appl., Springer Verlag, Berlin, 196211, 2001.
[3] J.M Harris, J.L.Hirst, M. Mossinghoff: Combinatorics and Graph Theory, Springer, New
York, 2008.
Department of Mathematics
University of Kwazulu-Natal
Howard College, Durban
E-mail address: winter@ukzn.ac.za


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

http://repository.research-publication.com/46/1/AMSJ-Vol3-N2-4.pdf

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

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