Advances in Mathematics: Scientific Journal 9 (2020), no.7, 4721–4727
ADV MATH
SCI JOURNAL ISSN: 1857-8365 (printed); 1857-8438 (electronic)
https://doi.org/10.37418/amsj.9.7.38
AVERAGE DETOUR D-DISTANCE IN GRAPHS
P. L. N. VARMA 1 , V. VENKATESWARA RAO, D. REDDY BABU, T. NAGESWARA RAO,
AND Y. SRINIVASA RAO
A BSTRACT. The average distance is one of the important parameters in graph
theory. This article deals with the average distance between vertices using detour D-distance. We obtain some results by comparing the average detour Ddistance of two graphs. Further, we work out the average detour D-distance of
some families of graphs.
1. I NTRODUCTION
The average distance is most significant parameters in graph theory. The
applications of average distance are networks or in general good connection
of networks. It is used as a tool of analytic network where the performance
time is proportional to the distance between two points. In [2], Reddy Babu
and Varma have introduced the representation of D-distance and extended to
average D-distance between vertices in [3]. In previous article, the first two
authors have introduced the idea of detour D-distance and some work related,
see [4,5]. In present article we investigate the average D-distance between
vertices using detour distance. Next, in section 2, we obtain some results by
comparing the average detour D-distances of two graphs. In section 3, we work
out the average detour D-distance of various families of graphs. In any connected graph contains n vertices, the total detour D-distance (abbreviated as
1
corresponding author
2010 Mathematics Subject Classification. 05C12.
Key words and phrases. Detour distance, Average Detour distance, Detour diameter.
4721
4722
P. L. N. VARMA, V. V. RAO, D. REDDY BABU, T. N. RAO, AND Y. S. RAO
TDDD)is the addition of the detour distances among all possible pairs of verP
tices, i.e.,T DDD = u,v DD (u, v). The average detour D-distance is given by
1 T DDD
T DDD
). The detour D-distance representation as a matrix of
µD
D = n(n−1) = 2 ( nC
2
of the graph representation by MDD (G), as MDD (G) = [DD (ui , uj )]n×n where
DD (ui , uj ) is the detour D-distance between the vertices ui and uj . Clearly,
the detour D-distance matrix is symmetric matrix with order n × n.All diagonal entices being zero. The average degree of the graph is given by d(G) = |V1 |
P
deg(v) where means degree of the in graph . Throughout this article, graph
vǫV
mean simple, connected, finite and undirected graph. For any inexplicable notations and terms we refer [1].
2. R ESULTS
ON AVERAGE DETOUR
D- DISTANCE
We begin with a theorem and later give some consequences.
Theorem 2.1. Let two graphs G1 and G2 having the same number of vertices and
D
D
D
diamD
D (G1 ) < diamD (G2 ). If |E1 | < |E2 | then µD (G1 ) < µD (G2 ).
D
Proof. Since diamD
D (G1 ) < diamD (G2 ), the biggest entry in the detour D-matrix
of G1 is less than G2 and this causes total detour D-distance is to increase. Because the same order and the number of edges of G1 is less than the number of
D
edges of G2 , hence µD
D (G1 ) < µD (G2 ).
Theorem 2.2. Let two graphs G1 and G2 having the same number of vertices and
D
D
D
diamD
D (G1 ) < diamD (G2 ).if δ(G1 ) < δ(G2 ) then µD (G1 ) < µD (G2 )
Proof. Let G1 and G2 be two graphs having the same number of vertices and
D
diamD
D (G1 ) < diamD (G2 ). Then clearly δ(G1 ) < δ(G2 ) ⇒ |E1 | < |E2 | then from
D
Theorem 2.1, µD
D (G1 ) < µD (G2 ).
Theorem 2.3. Let two graphs G1 and G2 having the same number of vertices and
D
diamD
D (G1 ) < diamD (G2 ). If the mean degree of G1 is less than mean degree of G2
D
then µD
D (G1 ) < µD (G2 )
Proof. Let G1 and G2 be two graphs having the same number of vertices and
P
1
D
deg(v) = 21 deg(G).As
diamD
D (G1 ) < diamD (G2 ).We have by definition, |E| = 2
the graphs have same number of vertices and mean degree of G1 is less than
AVERAGE DETOUR D-DISTANCE IN GRAPHS
4723
D
D
mean degree of G2 we have degD
(G1 ) < degD
(G2 ) and hence |E1 | < |E2 |. Thus
D
by Th 2.1, hence we conclude that µD (G1 ) < µD
D (G2 ).
D
Theorem 2.4. Let S is a spanning subgraph of G,µD
D (S) < µD (G).
Proof. Consider S be a spanning subgraph of G,Then S and G are same order
and |E(S)| < |E(G)|. Thus by Theorem 2.1, hence we conclude that µD
D (S) <
D
µD (G).
3. AVERAGE
DETOUR
D- DISTANCE
OF VARIOUS FAMILIES OF GRAPHS
In this section we compute the average detour D-distance of various families
of graphs. We start on with complete graph.
Theorem 3.1. The average detour D-distance of Kn is n2 − 1 .
Proof. In a complete graph, every vertex has n − 1 adjacent vertices. The detour D-distance between every pair of vertices is n2 − 1. Thus the total detour
D-distance (TDDD) is 2(nC2 )(n2 − 1). Hence the average detour D-distance,
1 T DDD
) = n2 − 1.
µD
D (Kn ) = 2 ( nC
2
Next we compute the detour D-distance of a wheel graph.
Theorem 3.2. In a wheel graph, the average detour D-distance is 5n .
Proof. Consider the wheel graph, W1,n , on n+1 vertices {v0 , v1 , v2 , ..., vn }.Assume
that, without loss of generality, v0 is adjacent to all other vertices. Then degree of
and degree of v0 = n and degree of all other vertices is 3. The detour D-distance
between any pair of vertices is 5n. Thus the total detour D-distance (TDDD) is
2(n + 1C2 )(5n).
Hence
the
average
detour
D-distance,
1 T DDD
D
µD (W1,n ) = 2 ( n+1C ) = 5n.
2
Next we consider cyclic graph.
Theorem
3.3. In cyclic graph, the average detour D-distance is
9n2 −4n+8 if n is even
4(n−1)
µD
D (Cn )=
9n+5
if n is odd
4
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P. L. N. VARMA, V. V. RAO, D. REDDY BABU, T. N. RAO, AND Y. S. RAO
Proof. In a cyclic graph Cn , with vertices, each vertex has two adjacent vertices.
We consider independently cases if neven and odd.
Case 1: n is even Detour D-distances between pairs of vertices are as shown
below:
v1
v2
v3
... v n2 −1
v n2
v n2 +1 v n2 +2 ... vn−1
vn
3n+10
3n+4
3n+10
v1
0
3n − 1 3n − 4 ... 3n+16
... 3n − 4 3n − 1
2
2
2
2
3n+22
3n+16
3n+4
3n+4
v2 3n − 1
0
3n − 1 ...
... 3n − 7 3n − 4
2
2
2
2
3n+28
3n+22
3n+16
3n+10
v3 3n − 4 3n − 1
0
...
... 3n − 10 3n − 7
2
2
2
2
..
..
..
..
..
..
..
..
..
..
..
..
.
.
.
.
.
.
.
.
.
.
.
.
v n2 −1
v n2
v n2 +1
v n2 +2
..
.
vn−1
vn
3n+16
2
3n+10
2
3n+4
2
3n+10
2
3n+22
2
3n+16
2
3n+10
2
3n+4
2
3n+28
2
3n+22
2
3n+16
2
3n+10
2
3n+10
...
0
3n − 1 3n − 4 3n − 7 ... 3n+4
2
2
3n+4
... 3n − 1
0
3n − 1 3n − 4 ... 3n+10
2
2
3n+10
... 3n − 4 3n − 1
0
3n − 1 ... 3n+16
2
2
3n+16
... 3n − 7 3n − 4 3n − 1
0
... 3n+22
2
2
..
..
..
..
..
..
..
..
..
..
..
.
.
.
.
.
.
.
.
.
.
.
3n+4
3n+10
3n+16
3n+22
3n − 4 3n − 7 3n − 10 ...
...
0
3n − 1
2
2
2
2
3n+10
3n+4
3n+10
3n+16
3n − 1 3n − 4 3n − 7 ...
... 3n − 1
0
2
2
2
2
Table: 1 Detour D-distance of cyclic graphs ( n is even)
v1
v2
v3
v1
0
3n − 1 3n − 4
v2 3n − 1
0
3n − 1
v3 3n − 4 3n − 1
0
..
..
..
..
.
.
.
.
v n−3
2
v n−1
2
v n+1
2
v n+3
2
..
.
vn−1
vn
3n+13
2
3n+7
2
3n+7
2
3n+13
2
3n+19
2
3n+13
2
3n+7
2
3n+7
2
3n+25
2
3n+19
2
3n+13
2
3n+7
2
... v n−3
2
...
...
...
..
.
...
3n+13
2
3n+19
2
3n+25
2
..
.
0
... 3n − 1
v n−1
v n+1
v n+3 ...
3n+7
2
3n+13
2
3n+19
2
3n+7
2
3n+7
2
3n+13
2
3n+13
2
3n+7
2
3n+7
2
2
2
2
vn−1
vn
... 3n − 4 3n − 1
... 3n − 7 3n − 4
... 3n − 10 3n − 7
..
..
..
..
..
..
.
.
.
.
.
.
3n+7
3n+7
3n − 1 3n − 4 3n − 7 ...
2
2
0
... 3n − 4 3n − 1
3n − 1 3n − 4 ...
0
... 3n − 7 3n − 4 3n − 1
..
..
..
..
..
..
..
.
.
.
.
.
.
.
3n+7
3n+13
3n+19
3n − 4 3n − 7 3n − 10 ...
2
2
2
3n+7
3n+13
3n − 1 3n − 1 3n − 7 ... 3n+7
2
2
2
3n − 1 ...
0
..
.
3n+25
2
...
..
.
3n+13
2
3n+19
2
3n+25
2
..
.
... 3n − 1
... 3n − 1
Table: 2 Detour D-distance of cyclic graphs ( n is odd)
The sumn of the elements in each row is
( − 1)
n
3n + 4
[2(3n − 1) + ( − 1 − 1)(−3)] + (
)
Sn = 2 2
2
2
2
3n+7
2
3n+13
2
3n+19
2
..
.
0
0
AVERAGE DETOUR D-DISTANCE IN GRAPHS
4725
3n + 4
9n2 − 4n − 8
n−2
(9n + 8) + (
)=
. There are n number of rows. Thus
4
2
4
n(9n2 − 4n − 8)
.
the total detour D-distance is T DDD =
4
(9n2 − 4n + 8)
1 T DDD
Hence the average detour D-distance µD
)
=
(C
)
=
(
.
n
D
2 n C2
4(n − 1)
Case 2: n is odd
Detours D-distance between pairs of vertices are as shown in table 2.
The sum of the elements in each row is
=
Sn = [
( n2 − 1)
n
n − 1 (9n + 5)
(9n + 5)(n − 1)
[2(3n−1)+( −1−1)(−3)]] = [
]=
.
2
2
4
2
8
There are n number of rows. Thus total detour D-distance, TDDD, is
n(n − 1)(9n + 5)
. Hence the average detour D-distance
n × Sn =
4
1 T DDD
(9n + 5)
µD
)=
.
D (Cn ) = (
2 n C2
4
Next we go through complete bipartite graph.
Theorem 3.4. Let the graph be a complete bipartite graph,Km,n (m < n), the average D-distance is
µD
D (Km,n ) =
n(n−1)(m2 +mn+3n)+2mn(m2 +mn+2m−1)+m(m−1)(m2 +mn+m−2)
(m+n)(m+n−1)
.
Proof. The partition of the two vertex set of Km,n be able representation as
A ∪ B, where A = {v1 , v2 , v3 , ..., vm },B = {w1 , w2 , w3 , ..., wn }. Then DD (vi , vj ) =
m2 + mn + 3m,DD (wi , wj ) = m2 + mn + m − 2,DD (vi , wj ) = m2 + mn + 2m − 1 ,
see [4]. Thus the total detour D-distance is T DDD = nC2 (m2 + mn + 3n) +
mn(m2 + 2m + mn − 1) + mC2 (m2 + mn + m − 2).
Hence the average detour D-distance
1 T DDD
(T DDD)
)=
µD
D (Km,n ) = (
2 (m + n)C2
(m + n)(m + n − 1)
2
n(n − 1)(m + mn + 3n) + 2mn(m2 + mn + 2m − 1) + m(m − 1)(m2 + mn + m − 2)
=
.
(m + n)(m + n − 1)
Theorem 3.5. The average detour D-distance of,Km,m , is
µD
D (Km,m ) =
4m3 + m2 − 4m + 2
.
2m − 1
4726
P. L. N. VARMA, V. V. RAO, D. REDDY BABU, T. N. RAO, AND Y. S. RAO
Proof. Let Km,m be a complete bipartite graph. The vertex set of Km,m ,can
be written as A ∪ B, where A = {v1 , v2 , v3 , ..., vm },B = {w1 , w2 , w3 , ..., wm }.In
the complete bipartite graph the detour D-distances between different pairs are
DD (vi , vj ) = 2m2 +m−2,DD (vi , wj ) = 2m2 +2m−1. The total detour D-distance
(TDDD) is twice the mC2 (2m2 + m − 2) + m2 (2m2 + 2m − 1).
Hence
1 T DDD
)
(
µD
D (Km,n ) =
2 2mC2
4m3 + m2 − 4m + 2
mC2 (2m2 + m − 2) + m2 (2m2 + m − 1)
=
.
=
(2m)(2m − 1)
(2m − 1)
Now we consider graphs which are trees.
Theorem 3.6. The average detour D-distance of the path graph is µD
D (Pn ) =
where an = an−1 + n + 1 with a1 = 0 .
2an
,
n
Proof. Let Pn ,the detour D-distance between two vertices is same as the Ddistance as there is a single pathway connecting any two vertices. Thus the
outcome from Theorem 4.4 in [3].
Theorem
3.7. In a star graph, the average detour D-distance is
2(n + 2) + (n − 1)(n + 4)
.
µD
D (St1,n ) =
n−1
Proof. In a star graph St1,n ,the detour D-distance is same as the D-distance. Thus
the result follows from theorem 4.5 of [3].
R EFERENCES
[1] F. H ARARY: Graph Theory, Addison Wesley, 1969.
[2] D. R EDDY B ABU , P. L. N. VARMA: Distance in graphs, Gold. Res. Thoughts, 2 (2013),
53–58.
[3] D. R EDDY B ABU , P. L. N. VARMA: Average D-distance between vertices of a graph, Italian
Journal of Pure and Applied Mathematics , 33( 2014), 293–298.
[4] V. V ENKATESWARA R AO, P. L. N. VARMA: Detour Distance in Graphs w.r.t. D-distance,
Ponte International Journal of Sciences and Research. Res., 73(7) (2017),19–28.
[5] V. V ENKATESWARA R AO, D. R EDDY B ABU , P. L. N. VARMA: Distance in graphs-taking
thelong view, AKCE J. Graphs combin., 1 (2004), 1–13.
AVERAGE DETOUR D-DISTANCE IN GRAPHS
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