Steiner Degree Distance of Two Graph Products

Steiner Degree Distance of Two Graph Products

Yaping Mao, Zhao Wang, Kinkar Ch. Das

Abstract

The degree distanceDD(G) of a connected graphGwas invented by Dobrynin and Kochetova in 1994. Recently, one of the present authors introduced the concept ofk-center Steiner degree distance defined as

SDDk(G) = X

S⊆V(G)

|S|=k

"

X

v∈S

degG(v)

# dG(S),

where dG(S) is the Steiner k-distance of S and degG(v) is the degree of the vertexv inG. In this paper, we investigate the Steiner degree distance of complete and Cartesian product graphs.

1 Introduction

All graphs in this paper are assumed to be undirected, finite, and simple.

LetG be such a graph with vertex set V(G) and edge set E(G). Then the order and size of G are n = n(G) = |V(G)| and m = m(G) = |E(G)|. In

Key Words: Steiner degree distance, Degree distance, Complete product, Cartesian product.

2010 Mathematics Subject Classification: Primary 05C07; Secondary 05C90.

Corresponding author (Kinkar Ch. Das) Received: 10.04.2018.

Accepted: 30.07. 2018

83

other words,Ghasnvertices andmedges. The degreedegG(u) of the vertex u∈V(G) is number of first neighbors of this vertex.

Distance is one of the basic concepts of graph theory [7]. IfGis connected andu, v∈V(G), then the distance d(u, v) =dG(u, v) betweenuand v is the length of a shortest path connectinguandv. Ifv is a vertex of a connected graphG, then theeccentricity ε(v) ofv is defined asε(v) = max{d(u, v)|u∈ V(G)}. Furthermore, the radius rad(G) and diameter diam(G) of G are rad(G) = min{ε(v)|v∈V(G)}and diam(G) = max{ε(v)|v∈V(G)}. These latter two quantities are related by the inequalities rad(G) ≤ diam(G) ≤ 2rad(G). More details on this subject can be found in [14]. We refer to [5]

for graph theoretical notation and terminology not specified here.

For a graphGwith vertex setV(G), thedegree distance is defined as [13]

DD=DD(G) = X

{u,v}∈V(G)

[deg_G(u) +deg_G(v)]d_G(u, v). (1) For more details on degree–and–distance–based graph invariant, we refer to [2, 3, 4, 6, 17, 18, 30, 32, 35].

The Wiener indexW(G) of the graphGis defined as

W(G) = X

{u,v}⊆V(G)

dG(u, v). (2)

Details on this oldest distance–based topological index can be found in nu- merous surveys, e.g., in [11, 12, 19, 20, 33, 34, 36].

The Steiner distance of a graph, introduced by Chartrand et al. [9] in 1989, is a natural and nice generalization of the concept of classical graph distance.

For a subset S of the vertex set V(G), consisting of at least two vertices, theSteiner distance d(S) (or simply the distance of S) is the minimum size (number of edges) of a connected subgraphs whose vertex set containsS. This connected subgraph is necessarily a tree and is referred to as aSteiner tree.

Note that ifS={u, v}, thend(S) =d(u, v) is nothing new, but the classical distance betweenuandv. Clearly, if|S|=k, thend(S)≥k−1.

Let nand k be integers such that 2≤k≤n. TheSteiner k-eccentricity εk(v) of a vertexv ofGisεk(v) = max{d(S)|S⊆V(G), |S|=k,andv∈S}.

TheSteiner k-radius ofGis sradk(G) = min{εk(v)|v∈V(G)}, whereas the Steiner k-diameter of G is sdiam_k(G) = max{ε_k(v)|v ∈ V(G)}. Note that for every connected graphG,ε₂(v) =ε(v) for all verticesv ofG,srad₂(G) = rad(G) andsdiam₂(G) =diam(G). For more details on Steiner distance, we refer to [1, 8, 9, 10, 14, 31].

The following result is immediate.

Observation 1.1. If H is a connected spanning subgraph ofG, then

sdiamk(G)≤sdiamk(H) holds for allk, 2≤k≤n.

Li et al. [21] put forward a Steiner–distance–based generalization of the Wiener index concept. According to [21], thek-center Steiner Wiener index SWk(G) of the graphGis defined by

SWk(G) = X

S⊆V(G)

|S|=k

d(S). (3)

Fork= 2, the above defined Steiner Wiener index coincides with the ordinary Wiener index, Eq. (2). It is usual to considerSW_k for 2 ≤k ≤n−1, but the above definition would be applicable also in the casesk = 1 and k=n, implying SW1(G) = 0 and SWn(G) = n−1. For more details on Steiner Wiener index, we refer to [21, 22, 26, 27, 28, 29].

Recently, Gutman [16] offered an analogous generalization of the concept of degree distance, Eq. (1). Thus, thek-center Steiner degree distanceSDDk(G)

ofGis defined as

SDDk(G) = X

S⊆V(G)

|S|=k

"

X

v∈S

degG(v)

#

dG(S). (4)

Mao et al. [28] reported expressions for thek-center Steiner degree distance of the star, path, as well as the complete and complete bipartite graphs, and got general expressions forSDD_k(G) for k = n, n−1. They also obtained sharp lower and upper bounds forSDD_k. Very recently, Mao and Das [25]

generalize the concept of Gutman index by Steiner distance. The Steiner Gutmank-indexSGutk(G) of graphGis defined by

SGutk(G) = X

S⊆V(G)

|S|=k

"

Y

v∈S

degG(v)

# dG(S).

The join and Cartesian products are defined as follows:

Thejoinorcomplete productG∨Hof two disjoint graphsGandH, is the graph with vertex setV(G)∪V(H) and edge setE(G)∪E(H)∪{uv|u∈ V(G), v∈V(H)}.

TheCartesian productGH of two graphsGandH, is the graph with vertex set V(G)×V(H), in which two vertices (u, v) and (u⁰, v⁰) are adjacent if and only ifu=u⁰ andvv⁰∈E(H), orv=v⁰anduu⁰∈E(G).

2 Complete product

In this section, we give the exact expression ofSDDk(G∨H).

Theorem 2.1. LetG, H be two connected graphs of ordera, b(a≤b), respec- tively. Letk be an integer with 3≤k≤a+b.

(1)If 1≤k≤a, then SDDk(G∨H) = k

a k

−x−2e(G) a−1

k−1

−

x

X

i=1

X

v∈S_i

degG(v)

!

+k b

k

−y−2e(H) b−1

k−1

−

y

X

i=1



 X

v∈S⁰_i

deg_H(v)





+2(k−1)[e(G) +e(H) +ab]

a+b−1 k−1

−(k−1)ab a−1

k−1

−(k−1)ab b−1

k−1

,

where S₁, S₂, . . . , S_x are all the k-subsets of V(G) such that G[S_i] (1 ≤ i ≤ x) is connected, and S⁰₁, S₂⁰, . . . , S_y⁰ are all the k-subsets of V(H) such that H[S⁰_i] (1≤i≤y)is connected.

(2)If a < k≤b, then SDDk(G∨H) = ak²

b k

−akx+ 2e(H) b−1

k−1

−

x

X

i=1

X

v∈Si

degH(v)

!

+2(k−1)[e(G) +e(H) +ab]

a+b−1 k−1

−(k−1)ab a−1

k−1

−(k−1)ab b−1

k−1

−(k−1)·2e(G) a−1

k−1

,

whereS1, S2, . . . , Sx are all thek-subsets ofV(H)such thatH[Si] (1≤i≤x) is connected.

(3)If b < k≤a+b, then SDDk(G∨H) = 2(k−1)e(G)

a+b−1 k−1

− a−1

k−1

−(k−1)ab

"

a−1 k−1

+ b−1

k−1 #

+ 2(k−1)e(H)

a+b−1 k−1

− b−1

k−1

.

Proof. Let V(G) = {u1, u2, . . . , ua} and V(H) = {v1, v2, . . . , vb}. Without loss of generality, we can assume that a ≤ b. We distinguish the following three cases to show this theorem.

Case 1. 1≤k≤a. For any S⊆V(G) and|S|=k, we haveS∩V(G) =∅, orS∩V(H) =∅, or S∩V(G)6=∅and S∩V(H)6=∅.

If S ∩V(H) = ∅, then S ⊆ V(G). Recall that S₁, S₂, . . . , S(^a_k) be all thek-subsets of V(G), andS₁, S₂, . . . , S_x are all the k-subsets of V(G) such that G[S_i] (1 ≤ i ≤x) is connected. Then d_G(S_i) = d_G∨H(S_i) = k−1 for eachi (1 ≤ i ≤x). For each i (x+ 1 ≤i ≤ _k^a

), we haved_G(S_i) ≥k and d_G∨H(S_i) =k. Forv∈V(G∨H), ifv∈V(G), thendeg_G∨H(v) =deg_G(v) +b.

This case contributes toSDDk(G∨H) by X

S⊆V(G)

|S|=k

X

v∈S

deg_G∨H(v)

!

d_G∨H(S)

= X

S⊆V(G)

|S|=k

X

v∈S

(degG(v) +b)

!

dG∨H(S)

= X

S⊆V(G)

|S|=k

X

v∈S

deg_G(v)

!

d_G∨H(S) +bk X

S⊆V(G)

|S|=k

d_G∨H(S)

= (k−1)X

S⊆V(G),|S|=k

dG(S)=k−1

X

v∈S

degG(v)

!

+kX

S⊆V(G), |S|=k

dG(S)≥k

X

v∈S

degG(v)

!

+bkX

S⊆V(G)

|S|=k

d_G∨H(S)

= k X

S⊆V(G)

|S|=k

X

v∈S

degG(v)

!

− X

S⊆V(G),|S|=k

d_G(S)=k−1

X

v∈S

degG(v)

! +bk

k

a k

−x

= 2ke(G) a−1

k−1

−

x

X

i=1

X

v∈Si

deg_G(v)

! +bk²

a k

−bkx.

IfS∩V(G) =∅, thenS⊆V(H). Similarly, this case contributes toSDDk(G∨

H) by

2ke(H) b−1

k−1

−

y

X

i=1



 X

v∈S_i⁰

degH(v)



+ak² b

k

−aky.

SupposeS∩V(G)6=∅and S∩V(H)6=∅. In this case, dG∨H(S) = k−1.

This case contributes toSDDk(G∨H) by X

S⊆V(G),|S|=k

S∩V(G)6=∅,S∩V(H)6=∅

X

v∈S

deg_G∨H(v)

!

d_G∨H(S)

= (k−1) X

S⊆V(G),|S|=k

S∩V(G)6=∅,S∩V(H)6=∅

X

v∈S

deg_G∨H(v)

!

= (k−1)·2e(G∨H)

a+b−1 k−1

−(k−1)·2e(G) a−1

k−1

−(k−1)ab a−1

k−1

−(k−1)·2e(H) b−1

k−1

−(k−1)ab b−1

k−1

= (k−1)(2e(G) + 2e(H) + 2ab)

a+b−1 k−1

−(k−1)·2e(G) a−1

k−1

−(k−1)ab a−1

k−1

−(k−1)·2e(H) b−1

k−1

−(k−1)ab b−1

k−1

.

From the above arguments, we conclude that SDDk(G∨H) = bk²

a k

−bkx+ 2e(G) a−1

k−1

−

x

X

i=1

X

v∈Si

degG(v)

!

+ak² b

k

−aky+ 2e(H) b−1

k−1

−

y

X

i=1



 X

v∈S_i⁰

degH(v)





+2(k−1)[e(G) +e(H) +ab]

a+b−1 k−1

−(k−1)ab a−1

k−1

−(k−1)ab b−1

k−1

,

where S1, S2, . . . , Sx are all the k-subsets ofV(G) such that G[Si] (1 ≤i ≤ x) is connected, and S₁⁰, S₂⁰, . . . , S_y⁰ are all the k-subsets of V(H) such that H[S⁰_i] (1≤i≤y) is connected.

Case 2. a < k≤b. For anyS⊆V(G) and|S|=k, we haveS∩V(G) =∅, orS∩V(G)6=∅andS∩V(H)6=∅.

If S ∩V(G) = ∅, then S ⊆ V(H). Recall that S₁, S₂, . . . , S(_k^b) be all thek-subsets ofV(H), and S₁, S₂, . . . , S_x are all the k-subsets ofV(H) such that G[S_i] (1 ≤i ≤ x) is connected. Then d_H(S_i) = d_G∨H(S_i) =k−1 for eachi (1 ≤ i≤ x). For each i (x+ 1 ≤i ≤ ^b_k

), we haved_H(S_i) ≥k and d_G∨H(S_i) =k. Forv∈V(G∨H), ifv∈V(H), thendeg_G∨H(v) =deg_H(v)+a.

This case contributes toSDDk(G∨H) by X

S⊆V(H)

|S|=k

X

v∈S

deg_G∨H(v)

!

d_G∨H(S)

= X

S⊆V(H)

|S|=k

X

v∈S

(degH(v) +a)

!

dG∨H(S)

= X

S⊆V(H)

|S|=k

X

v∈S

deg_H(v)

!

d_G∨H(S) +ak X

S⊆V(H)

|S|=k

d_G∨H(S)

= (k−1) X

S⊆V(H), |S|=k

dH(S)=k−1

X

v∈S

degH(v)

!

+k X

S⊆V(H), |S|=k

dH(S)≥k

X

v∈S

degH(v)

!

+ak X

S⊆V(H)

|S|=k

d_G∨H(S)

= k X

S⊆V(H)

|S|=k

X

v∈S

degH(v)

!

− X

S⊆V(G),|S|=k

dH(S)=k−1

X

v∈S

degH(v)

! +ak

k

b k

−x

= 2ke(H) b−1

k−1

−

x

X

i=1

X

v∈S_i

degH(v)

! +ak²

b k

−akx .

SupposeS∩V(G)6=∅and S∩V(H)6=∅. In this case, d_G∨H(S) = k−1.

Then this case contributes toSDD_k(G∨H) by X

S⊆V(G),|S|=k

S∩V(G)6=∅,S∩V(H)6=∅

X

v∈S

degG∨H(v)

!

dG∨H(S)

= (k−1) X

S⊆V(G),|S|=k

S∩V(G)6=∅,S∩V(H)6=∅

X

v∈S

deg_G∨H(v)

!

= (k−1)·2e(G∨H)

a+b−1 k−1

−(k−1)·2e(G) a−1

k−1

−(k−1)ab a−1

k−1

−(k−1)·2e(H) b−1

k−1

−(k−1)ab b−1

k−1

= (k−1)(2e(G) + 2e(H) + 2ab)

a+b−1 k−1

−(k−1)·2e(G) a−1

k−1

−(k−1)ab a−1

k−1

−(k−1)·2e(H) b−1

k−1

−(k−1)ab b−1

k−1

.

From the above arguments, we conclude that SDDk(G∨H) = ak²

b k

−akx+ 2e(H) b−1

k−1

−

x

X

i=1

X

v∈Si

degH(v)

!

+2(k−1)[e(G) +e(H) +ab]

a+b−1 k−1

−(k−1)ab a−1

k−1

−(k−1)ab b−1

k−1

−(k−1)·2e(G) a−1

k−1

,

whereS1, S2, . . . , Sxare all thek-subsets ofV(H) such thatH[Si] (1≤i≤x) is connected.

Case 3. b < k≤a+b. For anyS⊆V(G) and|S|=k, we haveS∩V(G)6=∅ andS∩V(H)6=∅. In this case, d_G∨H(S) =k−1. Then

SDDk(G∨H)

= X

S⊆V(G)

|S|=k

X

v∈S

deg_G∨H(v)

!

d_G∨H(S)

= (k−1) X

S⊆V(G)

|S|=k

X

v∈S

deg_G∨H(v)

!

= (k−1)·2e(G∨H)

a+b−1 k−1

−(k−1)·2e(G) a−1

k−1

−(k−1)ab a−1

k−1

−(k−1)·2e(H) b−1

k−1

−(k−1)ab b−1

k−1

= (k−1)[2e(G) + 2e(H) + 2ab]

a+b−1 k−1

−(k−1)·2e(G) a−1

k−1

−(k−1)ab a−1

k−1

−(k−1)·2e(H) b−1

k−1

−(k−1)ab b−1

k−1

=2(k−1)e(G)

a+b−1 k−1

− a−1

k−1

−(k−1)ab

a−1 k−1

+ b−1

k−1

+2(k−1)e(H)

a+b−1 k−1

− b−1

k−1

.

This completes the proof of the theorem.

2.1 Cartesian product

In [15], Gologranc obtained the following lower bound for Steienr distance.

Lemma 2.1. [15] Let k ≥ 2 be an integer, and let G, H be two connected graphs. Let S={(g1, h1),(g2, h2), . . . ,(gk, hk)}be a set of distinct vertices of

GH. LetSG={g1, g2, . . . , gk}andSH={h1, h2, . . . , hk}. Then d_GH(S)≥dG(SG) +dH(SH).

Mao et al. derived the following results for Steiner distance.

Lemma 2.2. Let G, H be two connected graphs of order n, m, respectively.

Letk be an integer with 3≤k≤mn. LetS={(g1, h1),(g2, h2), . . . ,(gk, hk)}

be a set of distinct vertices of GH. Let S_G = {g₁, g₂, . . . , g_k} and S_H = {h1, h2, . . . , hk}. Then

dG(SG) +dH(SH) ≤ d_GH(S)

≤ min{dG(SG) + (k−2)dH(SH), dH(SH) + (k−2)dG(SG)}.

In this section, we give the upper and lower bounds ofSDDk(GH).

Theorem 2.2. LetG, H be two connected graphs of ordera, b(a≤b), respec- tively. Letk be an integer with 3≤k≤a+b. Then

X≤SDDk(GH)≤ k−1

2

X,

where X=

k

X

i=1

b b−1 r1−1

! b r2

! . . . b

ri

!

SDDi(G) +

k

X

j=1

a a−1 s1−1

! a s2

! . . . a

sj

!

SDDj(H)

+2e(H)

k

X

i=1

b b−1 r1−1

! b r2

! . . . b

ri

!

iSWi(G)

+2e(G)

k

X

j=1

a a−1 s1−1

! a s2

! . . . a

sj

!

jSWj(H),

Pi

p=1rp=k,Pj

q=1sq =k,rp≥1 (1≤p≤i),sq≥1 (1≤q≤j).

Proof. From the definition of Steiner degree distance, we have SDDk(GH) = X

S⊆V(GH)

|S|=k



 X

(u,v)∈S

deg_GH(u, v)



d_GH(S).

LetS ={(u1, v1),(u2, v2), . . . ,(uk, vk)}. Then X

(u,v)∈S

deg_GH(u, v) =

k

X

i=1

deg_G(u_i) +

k

X

i=1

deg_H(v_i).

From Lemma 2.1, we havedGH(S)≥dG(SG) +dH(SH), and hence SDDk(GH)

≥ X

S⊆V(GH)

|S|=k k

X

i=1

degG(ui) +

k

X

i=1

degH(vi)

!

[dG(SG) +dH(SH)]

= X

S⊆V(GH)

|S|=k k

X

i=1

degG(ui)

!

dG(SG) + X

S⊆V(GH)

|S|=k k

X

i=1

degH(vi)

! dH(SH)

+ X

S⊆V(GH)

|S|=k k

X

i=1

degH(vi)

!

dG(SG) + X

S⊆V(GH)

|S|=k k

X

i=1

degG(ui)

! dH(SH)

=

k

X

i=1

b b−1 r1−1

! b r2

! . . . b

ri

!

SDDi(G) +

k

X

j=1

a a−1 s1−1

! a s2

! . . . a

sj

!

SDDj(H)

+2e(H)

k

X

i=1

b b−1 r1−1

! b r2

! . . . b

ri

!

iSWi(G)

+2e(G)

k

X

j=1

a a−1 s1−1

! a s2

! . . . a

sj

!

jSWj(H)

From Lemma 2.2, we haved_GH(S)≤min{dG(SG)+dH(SH)(k−2), dH(SH)+

dG(SG)(k−2)} ≤ d^k−1₂ e[dG(SG) +dH(SH)], and hence SDDk(GH)

≤

k−1 2

X

S⊆V(GH)

|S|=k k

X

i=1

deg_G(u_i) +

k

X

i=1

deg_H(v_i)

!

[d_G(S_G) +d_H(S_H)]

=

k−1 2

^k X

i=1

b b−1

r1−1 b

r2

. . .

b ri

SDDi(G) +k 2

k

X

j=1

a a−1

s1−1 a

s2

. . . a

s_j

SDDj(H) +ke(H)

k

X

i=1

b b−1

r₁−1 b

r₂

. . . b

r_i

iSWi(G)

+ke(G)

k

X

j=1

a a−1

s1−1 a

s2

. . .

a sj

jSW_j(H).

This completes the proof of the theorem.

Remark 2.1. For k = 3, the lower and upper bounds on SDDk(GH) are same(X)in Theorem 2.2.

3 Conclusion

The Steiner distance in a graph, introduced by Chartrand et al. is a natural generalization of the concept of classical graph distance. The degree distance seems to have been considered first by Dobrynin and Kochetova, and prac- tically at the same time by Gutman, who used a different name for it. The k-center Steiner degree distance introduced very recently. In this paper, we obtain thek-center Steiner degree distance of complete and Cartesian product graphs. Investigating the other graph products (corona product, composition, disjunction etc.) are our future work.

Acknowledgment. The authors are much grateful to the anonymous referee

for his valuable comments on our paper, which have considerably improved the presentation of this paper. The first author was supported by the National Science Foundation of China (Nos. 11601254, 11551001, 11161037, 11461054) and the Science Found of Qinghai Province (Nos. 2016-ZJ-948Q, and 2014- ZJ-907). The third author was supported by the Sungkyun research fund, Sungkyunkwan University, 2017, and National Research Foundation of the Korean government with grant No. 2017R1D1A1B03028642.

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Yaping Mao,

Department of Mathematics, Qinghai Normal University, Xining, Qinghai 810008, China.

Center for Mathematics and Interdisciplinary Sciences of Qinghai Province, Xining, Qinghai 810008, China

e-mail: [email protected] Zhao Wang,

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

e-mail: [email protected] Kinkar Ch. Das,

Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea.

Email:[email protected]