Bounds on the closeness centrality of a graph

BOUNDS ON THE CLOSENESS CENTRALITY OF A GRAPH arXiv:2204.11283v2 [math.CO] 18 Jul 2023 THOMAS BRITZ, XIN HU, ABDELLAH ISLAM AND HOPEIN CHRISTOFEN TANG Abstract. We present new values and bounds on the (normalised) closeness centrality C̄C of connected graphs and on its product ¯ lC̄C with the mean distance ¯ l of these graphs. Our main result presents the fundamental bounds 1 ≤ ¯ lC̄C < 2. The lower bound is tight and the upper bound is asymptotically tight. Combining the lower bound with known upper bounds on the mean distance, we find ten new lower bounds for the closeness centrality of graphs. We also present explicit expressions for C̄C and l̄C̄C for specific families of
graphs. Elegantly
and perhaps surprisingly, the asymptotic values nC̄C Pn and of nC̄C Ln both equal π, and the asymptotic limits of ¯ lC̄C for these families of graphs are both equal to π/3. We conjecture that the set of values l̄C̄C for all connected graphs is dense in the interval [1, 2). 1. Introduction Let G = (V, E) be a connected graph on |V | = n vertices. Defined in 1950 by Bavelas [5], the (normalised) closeness centrality of G is defined as the average of the (normalised) closeness centralities C̄C (v) for each vertex v in G: n−1 1X C̄C (v) where C̄C (v) = X C̄C (G) := n d(v, w) v∈V w∈V and where d(v, w) is the distance between v and w; that is, the shortest length of a path between v and w in G. The purpose of this paper is to provide values and bounds on C̄C := C̄C (G) and on the product ¯ lC̄C where ¯l := ¯l(G) is the mean distance of G, defined as X 1 ¯l(G) = d(u, v) . n(n − 1) u,v∈V In Section 2, Theorem 1 presents the fundamental and elegant new bounds 1 ≤ ¯lC̄C < 2. The lower bound is tight and the upper bound is asymptotically tight. Although these bounds are simple, they appear to have been overlooked in previous literature. They show that the closeness centrality C̄C (G) and the mean distance ¯l(G) cannot both be small nor both be large. These bounds are also useful: the lower bound combined with known upper bounds on ¯l(G), the yields ten new lower bounds for C̄C (G); see Corollary 2. In Section 3, we prove four new upper bounds on C̄C (G) that depend on different graph parameters; see Theorems 6, 9 and 10. In Section 4, we provide lower bounds for three infinite families of graphs, namely the selfcomplementary graphs, the 2-connected graphs, and trees; see Theorem 12. We also provide explicit expressions for the closeness centralities C̄C (v) for the vertices in twelve specific infinite families of graphs, as well as the closeness centralities C̄C (G) for ten of these graphs; see Proposition 13 and Corollary 14. For the remaining two families, namely the path graphs Pn and the ladder graphs Ln , we provide asymptotically tight lower and upper bounds; see 1 2 THOMAS BRITZ, XIN HU, ABDELLAH ISLAM AND HOPEIN CHRISTOFEN TANG Propositions 16 and 17. From these bounds, we deduce Corollary 18
which states the elegant
and perhaps surprising fact that the asymptotic limits of nC̄C Pn and of nC̄C Ln both equal π. In Section 5, we derive the precise values of the product ¯lC̄C for the twelve families of graphs in Section 4 as well as another family of graphs. Again, we find surprising asymptotic values, π including the value lim ¯lC̄C = for path and ladder graphs; see Corollary 22. n→∞ 3 The final Section 6 is dedicated to proving Theorem 1. To do so, we first prove the stronger upper bound ¯ lC̄C < 2(1 − 1/n)2 + 2/n2 for any connected graph G with n vertices; see Theorem 26. We also construct a family of graphs whose values ¯lC̄C lie arbitrarily close to 2; see Theorem 30. We conclude the article by conjecturing that the set of values ¯lC̄C for all finite connected graphs is dense in the interval between 1 and 2. 2. Two fundamental bounds and ten lower bounds Consider any connected graph G = (V, E) on n vertices and m edges. For brevity, we will write C̄C := C̄C (G) and ¯l := ¯ l(G). Our first main result are elegantly interlinked lower and upper bounds for ¯l and C̄C . Theorem 1. 1 ≤ ¯l C̄C < 2 . The lower bound is tight and the upper bound is asymptotically tight. This theorem will be proved in Section 6. The proof shows that G achieves equality ¯l C̄C = 1 exactly when the terms PC̄C (v) are identical for all vertices v; equivalently, this is when the so-called transmission w d(v, w) is independent of the vertex v. Graphs with this property were introduced by Handa [15] as transmission regular graphs and include cycles, complete graphs, and all other distance-regular graphs but which, as shown in [2, 3], are not necessarily regular. For more information on distance-regular graphs, see, for instance [1, 4]. ¯ Theorem 1 provides the lower bound ¯l ≥ C̄−1 C on the mean distance l. This complements ¯ the many upper bounds on l found in the literature; see, for instance [6, 8, 9, 13, 20, 22, 23]. Conversely, these upper bounds on ¯l may be combined with the general lower bound C̄C ≥ ¯ l−1 from Theorem 1 to provide new lower bounds on the closeness centrality C̄C , as described in the following corollary. Corollary 2. Let G = (V, E) be a connected graph on n vertices and m edges with minimum vertex degree δ, maximum vertex degree ∆ and diameter D = max d(u, v). Then u,v∈V (1) C̄C (2) C̄C (3) C̄C (4) C̄C (5) C̄C (6) C̄C −1 n − ∆ ∆ − 1 2(m − n + 1) − +1 ; ≥ (n − 1) n ∆ n(n − 1)
−1 ≥ 6n(n − 1) 3n(n − 4)D + 6n(n + 2) − 12(m + 1) − D 2 (D − 6) + h(D) ;  j n(n − 1) k−1 X ; ≥ (n − 1) Q2k (0) + n − 1 k : Qk (0)≤n−1
−1 ≥ (n − 1)n bn(n − 1) − 2(b − 1)m ;  m 1 −1 n − 1 l ∆ + θ2 ln(n − 1) + ; ≥ n 4θ2 2 2(m − n − 2) ≥ (n − 1) −1 ; θ2 + · · · + θn−1  BOUNDS ON THE CLOSENESS CENTRALITY OF A GRAPH 3 3 ; n+1   (n + 1)n(n − 1) − 2m −1 (8) C̄C ≥ n(n − 1) ; δ+1 (9) C̄C ≥ α−1 ; (10) C̄C ≥ D −1 where h(D) = 6 − D if n − D is odd and h(D) = 2D when n − D is even; θi is the ith largest eigenvalue of the Laplacian matrix for G; b is the number of these Laplacian eigenvalues; Qk is the k-alternating polynomial defined over them; and α is the independence number for G. The first two bounds in Corollary 2 follow from upper bounds on ¯l proved by Gago, Hurajová and Madaras [13, 14]; the third and fourth bounds follow from upper bounds on ¯ l by Rodriguez and Yebra [22]; and the last six bounds follow from upper bounds on ¯l by Mohar [20], Teranishi [23], Entringer, Jackson and Snyder [10], Beezer, Riegsecker, and Smith [6], Chung [8], and Doyle and Graver [9], respectively. The last three of these bounds are tight and are attained exactly when G is the complete graph Kn . (7) C̄C ≥ The lower bound in Theorem 1 was first discovered by the second author in his thesis [17], expressed in terms of the betweenness centrality of G. This centrality was defined in 1977 by Freeman [11] as X σst (v) 1X CB (v) where CB (v) = , C̄B := C̄B (G) = n σst v∈V s6=t where σst is the number of shortest paths between vertices s and t and where σst (v) is the number of such paths that pass through vertex v 6= s, t. The betweenness centrality C̄B and the mean distance ¯l are related bijectively via the following identity first observed by Gago [12]. Theorem 3. [12] C̄B = 1 (n − 1)(¯l − 1). 2 The bound in Theorem 1 thus reflects that the betweenness centrality C̄B and the closeness centrality C̄C can be interpreted as dual concepts, as noted in [7]. The upper bound in Theorem 1 can be used together with known lower bounds on ¯l to give upper bounds for C̄C . However, we will instead present upper bounds in the following section that do not rely on Theorem 1. 3. Upper bounds on closeness centrality In Section 2, we proved lower bounds on the closeness centrality C̄C (G) of a graph G = (V, E). In this section, we prove upper bounds on C̄C (G). To derive the first bound, we make a trivial but useful observation. Remark 4. If e is an edge of a connected graph G for which G − e is also connected, then C̄C (G − e) < C̄C (G). A complete graph is obtained by adding edges to G, and it is easy to see that complete graphs have closeness centrality C̄C (Kn ) = 1. Remark 4 therefore implies the following simple upper bound on C̄C (G). Lemma 5. Let G be a connected graph. Then C̄C (G) ≤ 1, and equality is attained exactly when G is a complete graph. We can improve Remark 4 by estimating the reduction of the closeness centrality. 4 THOMAS BRITZ, XIN HU, ABDELLAH ISLAM AND HOPEIN CHRISTOFEN TANG Theorem 6. Let G be a connected graph on n vertices and k edges. Suppose that e is an edge of G such that G − e is also connected. Then  2 2 2n − 2  . − C̄C (G − e) ≤ C̄C (G) − n (n − 1)(n + 2) − 2k (n − 1)(n + 2) + 1 − 2k Proof. Let u and v be the end-vertices of e. Then dG−e (u, v) ≥ dG (u, v) for all u, v ∈ V and dG−e (x, y) = dG−e (y, x) ≥ 2 = dG (x, y) + 1 = dG (y, x) + 1. Thus, C̄C (G − e) n/(n − 1) X 1 1 1 1 1 X X X + + ≤ −X −X dG (x, v) 1 + dG (y, v) u∈V dG (u, v) 1+ dG (x, v) dG (y, v) v∈V v∈V n = C̄C (G) − n−1 v∈V v∈V 1 1 X v∈V v∈V  +X   X X dG (x, v) 1 + dG (x, v) dG (y, v) 1 + dG (y, v) v∈V v∈V ! . v∈V 1 The proof follows from dG (w, v) ≤ (n−1)(n+2)−k for w ∈ {x, y} [10, Theorem 2.4].  2 v∈V X Remark 7. We can replace 12 (n − 1)(n + 2) − k with max dG (w, v) for particular classes X w∈V of graphs to obtain tighter bounds for those classes. v∈V Corollary 8. If G is a connected graph on n vertices and e edges, then 4(n − 1) 2
. C̄C (G) ≤ 1 − + n n (n − 1)(n + 2) − 2e
Proof. Obtain G by removing exactly n2 − e edges from Kn and apply Theorem 6.  The eccentricity ǫ(v) of a vertex v of a graph G is the greatest distance between v and any other vertex in G. The radius r(G) is the smallest eccentricity ǫ(v) of a vertex v of G. For each vertex v ∈ V and integer i = 1, . . . , ǫ(v), define di (v) = {u ∈ V : d(u, v) = i} . Theorem 9. Let G = (V, E) be a connected graph on n vertices with radius r ≥ 1. Then n−1
. C̄C (G) ≤ n − 1 + r2
Proof. If r = 1, then 2r = 0, so we may apply Lemma 5. Suppose then that r ≥ 2. Since the identity d1 (v) + · · · + dǫ(v) (v) = n − 1 is true for each vertex v ∈ V , the sum X (1) u∈V d(u, v) = ǫ(v) X idi (v) i=1 is minimal exactly when d1 (v) = n − ǫ(v) and di (v) = 1 for each i = 2, . . . , ǫ(v). Therefore, 1X n−1 n−1 1X 1X C̄C (v) ≤ = C̄C (G) =
. n n n − ǫ(v) + 2 + · · · + ǫ(v) n n − 1 + ǫ(v) v∈V v∈V Since ǫ(v) ≥ r for each vertex v ∈ V , the theorem follows. v∈V 2  The next theorem provides a similar bound to that in Theorem 9 but also takes into consideration the maximal vertex degree ∆ of G. BOUNDS ON THE CLOSENESS CENTRALITY OF A GRAPH 5 Theorem 10. Let G = (V, E) be a connected graph on n vertices with radius r ≥ 2 and maximum vertex degree ∆. Then n−1 . C̄C (G) ≤ 2n − 1 − ∆ + r(r − 3)/2 Proof. Since d1 (v) ≤ ∆ for each v ∈ V , the sum (1) is minimal when d1 (v) = ∆, d2 (v) = n − 1 − ∆ − (ǫ(v) − 2) and di (v) = 1 for each 3 ≤ i ≤ ǫ(v). Therefore, 1X n−1 1X C̄C (v) ≤ . C̄C (G) = n n 2n − 1 − ∆ + ǫ(v)(ǫ(v) − 3)/2 v∈V v∈V Since ǫ(v) ≥ r for each vertex v ∈ V , the theorem follows.  Remark 11. Note that the lower bounds on C̄C (G) in Corollary 2 can be combined with the upper bounds on C̄C (G) in Corollary 8 and Theorems 9 and 10 to form potentially interesting bounds of the form A ≤ B where A is one of the right-hand side expressions in the ten bounds of Corollary 2 and where B is one of the three expressions n−1 n−1 2 4n − 4
,
and 1− + . r n n (n − 1)(n + 2) − 2e 2n − 1 − ∆ + r(r − 3)/2 n−1+ 2 4. Closeness centralities for particular families of graphs The previous two sections presented lower and upper bounds on the closeness centrality C̄C (G) of a graph G. This section gives lower and upper bounds on C̄C (G) for particular classes of graphs. Lower bounds on C̄C are given for three particular classes of graphs in the following proposition. The bounds are attained by applying Theorem 1 to upper bounds on ¯l proved by Hendry [16], Plesnik [21], and Gago, Hujarová and Madaras [13], respectively. Theorem 12. Let G = (V, E) be a connected graph on n vertices. Then (1) if G is self-complementary, then (n ≡ 0 or 1 (mod 4) and)  8n − 8    13n − 12 , n ≡ 0 (mod 4) C̄C (G) ≥  8n   , n ≡ 1 (mod 4) ; 13n − 1 j 1 k−1 (2) if G is 2-vertex-connected or 2-edge-connected, then C̄C (G) ≥ (n − 1) n2 ; 4 n∆ (3) if T is a tree with maximal degree ∆, then C̄C (T ) ≥ . 2(n − ∆)(∆ − 1)(n − 1) + 2 The second bound is tight and is achieved when G is the cycle Cn . Bounds such as those in Theorem 12 and in the preceding sections are particularly useful if it is not possible to calculate a formula for the closeness centrality. For some classes of graphs, however, such calculations are possible. Proposition 13 below presents explicit expressions for the closeness centrality C̄C (v) for the vertices in twelve particular infinite families of graphs. The proofs of these expressions are easy and straightforward, and are therefore omitted. The expressions and their derivations first appeared in the second and third authors’ works [17, 18], along with expressions for the closeness and betweenness centralities of similar graphs. The twelve families of graphs also featured in [24] where their betweenness centralities were calculated. Each of the families is well-known, except possibly the cocktail party graph CP (n) 6 THOMAS BRITZ, XIN HU, ABDELLAH ISLAM AND HOPEIN CHRISTOFEN TANG and the crown graph Sn0 : these are the graphs obtained by deleting n vertex-disjoint edges from K2n and Kn,n , respectively. Proposition 13. The vertex closeness centralities C̄C (v) for all v ∈ V are given below for the families of graphs G = (V, E) shown in Figure 1. Complete graph Kn : Cycle graph Cn : Wheel graph Wn : Star graph Sn : Near-complete graph Kn − e: Cocktail party graph CP (n): C̄C (v) = 1 C̄C (v) = ⌊nn−1 2 /4⌋ n−1 C̄C (v) = 1 if v is the central vertex and 2n−5 otherwise n C̄C (v) = 1 if v is the central vertex and 2n−1 otherwise n−1 C̄C (v) = n if v is adjacent to e and 1 otherwise C̄C (v) = 2n−1 (2n m+k−1 if v ∈ {u1 , . . . , uk } Complete bipartite graph Km,k : C̄C (v) = m+2k−2 m+k−1 if v ∈ {v1 , . . . , vm } k+2m−2 Crown graph Sn0 : C̄C (v) = 2n−1 3n Path graph Pn : Ladder graph Ln : Circular ladder graph CLn : 4(n−1) C̄C (vk ) = (2k−n+1) 2 +n2 −1 for k = 0, . . . , n − 1 4n−2 C̄C (vk ) = C̄C (uk ) = (2k−n+1) 2 +n2 +2n−1 for k = 0, . . . , n − 1 n−1 C̄C (v) = 2⌊n2 /4⌋+n Hypercube graph Qk : C̄C (v) = Kn 2k −1 . k2k−1 Cn Wn u1 v1 Kn − e v0 v1 vn−2 vn−1 Pn CP (n) u0 u1 un−2 un−1 v0 v1 uk ··· v2 Sn ··· vm Km,k Sn0 CLn (≃ K2 × Cn ) Qk vn−2 vn−1 Ln (≃ K2 × Pn ) Figure 1. Twelve families of graphs The expressions for C̄C (v) in Proposition 13 can easily be used to find simple expressions for the closeness centrality C̄C (G) of ten of the twelve families of graphs G. BOUNDS ON THE CLOSENESS CENTRALITY OF A GRAPH 7 Corollary 14. C̄C (Kn ) C̄C (Cn ) =1 = ⌊nn−1 2 /4⌋ C̄C (CPn ) = C̄C (Km,k ) = C̄C (Wn ) = C̄C (Sn0 ) C̄C (Sn ) = C̄C (Kn − e) = n2 −4 n(2n−5) n2 +2n−1 2n2 +n−1 n2 −2 n2 = C̄C (CLn ) = C̄C (Qk ) = 2n−1 2n m+k−1 m m+k k+2m−2 2n−1 3n n−1 2⌊n2 /4⌋+n 2k −1 . k2k−1 + k m+2k−2
For the families of paths Pn and ladders Ln , closed-form expressions for the closeness centrality C̄C (G) can be found, for instance by using Maple [19]. Corollary 15. For the graphs G ∈ {Pn , Ln }, X abc  1 + an + bn̂  ψ C̄C (G) = − nn̂ 2 a,b=±1 √ where ψ(x) is the digamma function and where n̂ = √ n̂ = 1 − 2n − n2 and c = 2n−1 4n if G = Ln . 1 n 1 − n2 and c = if G = Pn , and Propositions 16 and 17 below provide approximations for C̄C (Pn ) and C̄C (Ln ) by way of lower and upper bounds which converge asymptotically as n grows large. Proposition 16. r r r  n−1 4 n−1 n−1 π n−1 −1 tan + n n . ≤ C̄C (Pn ) ≤ n n+1 n+1 n n + 1 n⌊ 2 ⌋⌈ 2 ⌉ Proof. By Proposition 13, n−1 X n C̄C (Pn ) = f (k) n−1 where f (x) = k=0 Since f is non-decreasing for all x ≤ n−1 2 and is symmetric in x = n ⌋−1 2 j n k ⌊ X n = C̄C (Pn ) − f f (k) + n−1 2 k=0 ≤ Z 4 . (2x − n + 1)2 + n2 − 1 n−1 X n−1 2 , f (k) ⌋+1 k=⌊ n 2 (n−1)/2 f (x)dx + 0 Z n−1 f (x)dx = Z n−1 f (x)dx . 0 (n−1)/2 Therefore by integration, j n k n ≤ C̄C (Pn ) − f n−1 2   2x − n + 1 # n−1   √ 2−1 4 n−1 n −1 √ √ =√ tan . n2 − 1 n2 − 1 n2 − 1 0 " 2 tan−1 Hence, n − 1 j n k 4 ≤ f C̄C (Pn ) − n 2 n r n−1 tan−1 n+1 r n−1 n+1  . 8 THOMAS BRITZ, XIN HU, ABDELLAH ISLAM AND HOPEIN CHRISTOFEN TANG Since r n−1 ≤ 1, n+1 π C̄C (Pn ) ≤ n r n − 1 n − 1 j n k π = + f n+1 n 2 n r n−1 n−1 + n n . n + 1 n⌊ 2 ⌋⌈ 2 ⌉ The lower bound is proved similarly.  The proof of Proposition 17 below is similar to that above. Proposition 17. r  
π 2n − 1 4n − 2 2 n−1 2 (2n − 1)2 −1 √ √ ≤ C̄C Ln ≤ + . tan + 3 2 n n + 2n − 1 n 2n + 5 n(2n + 5) n n2 + 2n − 1 n2 + 2n − 1 Interestingly, Propositions 16 and 17 imply that the closeness centrality measures for paths π and ladders both converge to as n grows large: n Corollary 18.

lim nC̄C Pn = lim nC̄C Ln = π . n→∞ n→∞ 5. Product of closeness centralities and mean distances for graph families By Theorem 1, the product of the closeness centrality and the mean distance of each graph G satisfies the inequality 1 ≤ ¯l C̄C < 2. In this section, we will calculate the product ¯ l C̄C explicitly for specific classes of graphs. To do so, we first present some easily-calculated mean distances ¯l(G). Proposition 19. The mean distances ¯l(G) for the families of graphs in Figure 1 are: ¯l(Kn ) = 1 ¯l(Cn ) = ⌊n2 /4⌋ n−1 ¯l(Wn ) = 2n−4 n ¯ l(Sn ) = ¯ l(Kn − e) = ¯ l(CPn ) = 2n n+1 n(n−1)+2 n(n−1) 2n 2n−1 ¯l(Km,k ) = ¯l(S 0 ) = n ¯l(Pn ) = m(k+2m−2)+k(m+2k−2) (m+k)(m+k−1) 3n 2n−1 n+1 3 ¯l(Ln ) = ¯l(CLn ) = ¯l(Qk ) = n+2 3 2⌊n2 /4⌋+n n−1 k2k−1 . 2k −1 For several well-known families of graphs, the values of ¯l C̄C attain the lower bound 1 in Theorem 1, as the following corollary to Proposition 19 shows. Corollary 20. ¯l C̄C = 1 for G ∈ {Kn , Cn , CPn , Sn0 , CLn , Qk }. For other families, ¯ l C̄C is always greater than 1 but converges towards 1 as n grows large. Corollary 21. lim ¯l C̄C = 1 for G ∈ {Sn , Kn − e, Wn }. n→∞ Meanwhile, the values of ¯l C̄C converge to unexpectedly interesting constants for path and ladder graphs. π Corollary 22. lim ¯l C̄C = for G ∈ {Pn , Ln }. n→∞ 3 Proof. Straightforward from Corollary 18 and Proposition 19.  BOUNDS ON THE CLOSENESS CENTRALITY OF A GRAPH Corollary 23. For G = Kn−k,k , 9 √ max ¯l C̄C ≤ 18 − 12 2 . lim n→∞ 1≤k≤n−1 Indeed, √ lim ¯lC̄C = 18 − 12 2 n→∞ for k= $ n − 2 q p % (3n − 4)2 − 2(3n − 4) 2(n − 1)(n − 2) 2 . Proof. Apply simple calculus and Maple [19] to Corollary 14 and Proposition 19.  We also consider another family of graphs that provides another unexpectedly interesting asymptotic limit. For any positive integer n, k, let Gn,k denote the graph obtained by attaching a path of k vertices to each vertex of the complete graph Kn ; see Figure 2 for an example. Figure 2. G5,2 Proposition 24. 2 2 4 2
¯l Gn,k = (k + 1) n − 3 k − 3 k − 1 (k + 1)n − 1 and k
(k + 1)n − 1 X 2 . C̄C Gn,k = k+1 j(j + 1) + (k − j)(k − j + 1) + (n − 1)(k + 1)(2j + k + 2) j=0
Sn i i Proof. Let Gn,k be written as Pk+1 = v0i v1i . . . vki is a path and E = + E where Pk+1 i=1  i j {v0 , v0 } : i, j ∈ {1, . . . , n}, i 6= j . For any i ∈ {1, . . . , n} and j ∈ {0, . . . , k}, X v∈V j−1 k k X
XX


X d vji , vri + d vji , vri + d vji , vrs d vji , v = r=1 = j−1 X r=1 r=j+1 r+ k X (r − j) + (n − 1) r=j+1 s6=i r=0 k X (j + r + 1) r=0 j(j + 1) (k − j)(k − j + 1) (n − 1)(k + 1)(2j + k + 2) = + + . 2 2 2 The values of ¯ l and C̄C can now be obtained directly from their definition.  10 THOMAS BRITZ, XIN HU, ABDELLAH ISLAM AND HOPEIN CHRISTOFEN TANG  Theorem 25. Let G = Gn,k . For each integer k ≥ 1, ln 3− Moreover, lim k→∞  lim ¯l C̄C = ln 3 .  n→∞ Proof. It is easy to check that lim ¯l C̄C = n→∞ inequalities Z k+1 f (x) dx ≤ 0 k X j=0 f (j) ≤ Z k X j=0 k+1 0  2 2  ≤ lim ¯l C̄C ≤ ln 3+ . n→∞ k+2 k 1 . The bounds follow from the 2j + k + 2 f (x − 1) dx where f (x) = 1 . 2x + k + 2  6. Proof of Theorem 1 In order to prove Theorem 1, we first prove the lower bound therein. Theorem 26. Let G be a connected graph. Then ¯l C̄C ≥ 1 . Proof. Note that X v∈V 1 1 XX = d(u, v) = n¯l , n−1 C̄C (v) v∈V u∈V so, by the Arithmetic-Harmonic Inequality, X C̄C (v) n v∈V ¯l C̄C = ¯l ≥ ¯l X = 1. 1 n C̄C (v) v∈V  Next, we prove an upper bound on ¯l C̄C that is stronger than that in Theorem 1. Theorem 27. Let G be a connected graph on n vertices. Then 2  ¯l C̄C < 2 1 − 1 + 2 . n n2 Proof. Since d(x, u) + d(u, y) ≥ d(x, y) for all x, y, u ∈ V , it holds that, for each u ∈ V , X X d(x, u) + d(u, y) ≥ d(x, y) . x,y∈V −{u} , x6=y Adding P x∈V d(x, u) + P x,y∈V −{u} d(u, y) to both sides yields a nice inequality for each u ∈ V : X X (2n − 2) d(u, v) ≥ d(x, y) . y∈V v∈V Finally, ¯ l C̄C is bounded as follows: X d(x, y) X 1 X 2 x,y∈V ¯l C̄C = 1 X + = n2 n n2 d(u, v) u∈V u∈V v∈V x,y∈V X d(x, y) x,y∈V −{u} X v∈V d(u, v) ≤ 2 2n − + n n2 X X d(x, y) 2 u∈V x,y∈V −{u} X x,y∈V d(x, y) . BOUNDS ON THE CLOSENESS CENTRALITY OF A GRAPH 11 Since equality is not possible here, and since X X d(x, y) = (n − 2) u∈V x,y∈V −{u} X d(x, y) , x,y∈V the proof follows.  Corollary 28. Let G be a connected graph. Then ¯l C̄C < 2 . Next, we consider one more class of graphs, as follows. For all positive integers k, m, n, let Hk,m,n be the graph obtained by joining edges between all vertices of the complete graph Kn to one endpoint of each of m copies of the path Pk+1 ; see Figure 3 for an example. Figure 3. H3,2,5 When m = 1, this family coincides with the graphs mentioned by Entringer, Jackson and

; Snyder [10, Figure 2.1]. Below are semi-explicit expressions for ¯l Hk,m,n and C̄C Hk,m,n
Sm i + E, ∪ K we omit the messy but easily-derived explicit forms. Write Hk,m,n as P n i=1 k+1  i = v0i v1i . . . vki is a path and E = {v0i , v} : i ∈ {1, . . . , m}, v ∈ Kn . where Pk+1 Proposition 29. Let V = V (Hk,m,n ). Then
P P
2n(n − 1) + nm(k + 1)(k + 2) + 2m kj=0 v∈V d vj1 , v ¯l Hk,m,n =

2 (k + 1)m + n (k + 1)m + n − 1 C̄C Hk,m,n
(k + 1)m + n − 1) = (k + 1)m + n) Proof. For each u ∈ V (Kn ), X v∈V d(u, v) = X d(u, v) + v∈V (Kn ) k X m 2n
P + 1 2n − 2 + m(k + 1)(k + 2) v∈V d vj , v j=0 k m X X i=1 j=0 k X
(j + 1) d v, vji = n − 1 + m j=0 =n−1+ m(k + 1)(k + 2) . 2 ! . 12 THOMAS BRITZ, XIN HU, ABDELLAH ISLAM AND HOPEIN CHRISTOFEN TANG For any i ∈ {1, . . . , m} and j ∈ {0, . . . , k}, we have X v∈V
d vji , v = = X v∈V (Kn ) X j−1 k k X
XX

X
d vji , vri + d vji , vri + d vji , v + d vji , vrs r=1 (j + 1) + j−1 X r=1 v∈V (Kn ) s6=i r=0 r=j+1 k X r+ (r − j) + (m − 1) r=j+1 k X (j + r + 2) r=0 j(j + 1) (k − j)(k − j + 1) (m − 1)(k + 1)(2j + k + 4) + + . 2 2 2 has (k + 1)m + n vertices, the values of ¯l and C̄C follow from their definition.  = (n − 1)(j + 1) + Since Hk,m,n Consider the examples below, calculated by using Maple [19]: H10,2,80 : l¯C̄C ≈ 1.2688 , H104 ,2,2×106 : ¯l C̄C ≈ 1.9547 , H106 ,3,3×109 : ¯l C̄C ≈ 1.9956 . These values suggests that, for some large values of n and k, the value of ¯l C̄C for some members of this family will get very close to 2. Indeed, this is true, as the following theorem shows. Theorem 30. For G = Hr2 ,1,r3 , lim ¯l C̄C = 2 . r→∞ Proof. By Proposition 29, 5 4 3 2
¯l Hr2,1,r3 = 3r + 4r + 9r + 3r + 3r + 2 3 2 3(r + r + 1)(r + 1) and C̄C Hr2 ,1,r3 Since k X j=0
r 2 (r + 1) = 3 r + r2 + 1 k X 2r + 2 r + 2r + 3 j=0 j + 1 j+
r 3 −r 2 2 2 − r2 4 r4 1
r 3 −r 2 2 2 − r2 4
r 4 − 2r 3 − r 2 − 4r − 2 ! .
is always positive, − 2r 3 − r 2 − 4r − 2

2 2r ¯l C̄C Hr2 ,1,r3 > ¯l Hr2 ,1,r3 r (r + 1) . r 3 + r 2 + 1 r 2 + 2r + 3
r 2 (r + 1) 2r The proof now follows since lim ¯l Hr2 ,1,r3 3 = 2 and since ¯l C̄C < 2 by 2 2 r→∞ r + r + 1 r + 2r + 3 Corollary 28.  We are now able to prove Theorem 1. Proof of Theorem 1. Apply Theorems 26, 27 and 30 and Corollary 20.  Finally, note that Theorem 30 has the following corollary. Corollary 31. For all ǫ > 0, there exists a connected graph G for which ¯l C̄C > 2 − ǫ. Partly inspired by this corollary, we suggest the following conjecture. Conjecture 32. The numbers ¯l C̄C for all connected graphs G form a dense subset of the interval [1, 2). BOUNDS ON THE CLOSENESS CENTRALITY OF A GRAPH 13 ORCID Thomas Britz: Xin Hu: Abdellah Islam: Hopein Christofen Tang: https://orcid.org/0000-0003-4891-3055 https://orcid.org/0000-0001-5733-4544 https://orcid.org/0000-0002-9659-299X https://orcid.org/0000-0003-1707-851X Declaration of competing interest There is no conflict of interest. References [1] A. Abiad, B. Brimkov, A. Erey, L. Leshock, X. Martı́nez-Rivera, S. O, S.-Y. Song, and J. Williford, On the Wiener index, distance cospectrality and transmission-regular graphs, Discrete Appl. Math. 230 (2017), 1–10. [2] M. Aouchiche, G. Caporossi and P. Hansen, Variable neighborhood search for extremal graphs. 8. Variations on Graffiti 105, Congr. Numer. 148 (2001), 129–144. [3] M. Aouchiche and P. Hansen, On a conjecture about the Szeged index, European J. Combin. 31 (2010), 1662–1666. [4] S. Barik, G. Sahoo, On the distance spectra of coronas, Linear Multilinear A. 65 (2017), 1617–1628. [5] A. Bavelas, Communication patterns in task oriented groups, J. Acoust. Soc. Amer. 57 (1950), 271–282. [6] R.A Beezer, R.A Riegsecker, and B.A. Smith, Using minimum degree to bound average distance, Discrete Math. 226 (2001), 365–371. [7] U. Brandes, S.P. Borgatti and L.C. Freeman, Maintaining the duality of closeness and betweenness centrality, Soc. Networks 44 (2016), 153–159. [8] F.R.K. Chung, The average distance and independence number, J. Graph Theory 12 (1988), 229–235. [9] J.K. Doyle and E. Graver, Mean distance in a graph, Discrete Math. 17 (1977), 147–154. [10] R.C. Entringer, D.E. Jackson, D.A. Snyder, Distance in graphs, Czechoslovak Math. J. 26 (1976), 283–296. [11] L.C. Freeman, A set of measures of centrality based on betweenness, Sociometry 40 (1977), 35–41. [12] S. Gago, Métodos espectrales y nuevas medidas modelos y parámetros en grafos pequeño-mundo invariantes deescala, Ph.D. Thesis, Universitat Politécnica de Catalunya, 2006. [13] S. Gago, J. Hurajová, and T. Madaras, Notes on the betweenness centrality of a graph, Math. Slovaca 62 (2012), 1–12. [14] S. Gago, J.C. Hurajová, and T. Madaras, On betweenness-uniform graphs, Czech. Math. J. 63 (2013), 629–642. [15] K. Handa, Bipartite graphs with balanced partitions, Ars Combin. 51 (1999), 113–119. [16] G.R.T. Hendry, On mean distance in certain classes of graphs, Networks 19 (1989), 451–457. [17] X. Hu, A Survey on Centrality Measure, Master’s thesis, UNSW Sydney, 2019. [18] A. Islam, Effect on Graph Operations on Graph Centrality Measures, MATH3001 Project thesis, UNSW Sydney, 2020. [19] Maplesoft, Maple 2020, Waterloo, Ontario, 2020. [20] B. Mohar, Eigenvalues, diameter, and mean distance in graphs. Graphs Combin. 7 (1991), 53–64. [21] J. Plesnı́k, On the sum of all distances in a graph or digraph, J. Graph Theory 8 (1984), 1–21. [22] J.A. Rodrı́guez and J.L.A. Yebra, Bounding the diameter and the mean distance of a graph from its eigenvalues: Laplacian versus adjacency matrix methods, Discrete Math. 196 (1999), 267–275. [23] Y. Teranishi, Laplacian spectra and invariants of graphs, Discrete Math. 257 (2002), 183–189. [24] S.K.R. Unnithan, B. Kannan, and M. Jathavedan, Betweenness centrality in some classes of graphs, Int. J. Comb. (2014), Art. ID 241723, 12 pp. [25] D. Vukičević and G. Caporossi, Network descriptors based on betweenness centrality and transmission and their extremal values, Discrete Appl. Math. 161 (2013), 2678–2686.