Average distance, radius and remoteness of a graph

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1 Also available at ISSN (printed edn.), ISSN (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (0) 5 Average distance, radius and remoteness of a graph Baoyindureng Wu College of Mathematics and System Science, Xinjiang University Urumqi 83006, P.R. China Wanping Zhang Foundation department, Karamay Vocational and Technical College Karamay , P.R. China Received 8 February 0, accepted 5 August 03, published online 6 December 03 Abstract Let G = (V, E) be a connected graph on n vertices. Denote by l(g) the average distance between all pairs of vertices in G. The remoteness ρ(g) of a connected graph G is the maximum average distance from a vertex of G to all others. The aim of this paper is to show that two conjectures in [5] concerned with average distance, radius and remoteness of a graph are true. Keywords: Distance, radius, eccentricity, proximity, remoteness. Math. Subj. Class.: 05C, 05C35 Introduction All graphs considered in this paper are finite and simple. For notation and terminology not defined here, we refer to West []. Let G = (V, E) be a finite simple graph with vertex set V and edge set E, V and E are its order and size, respectively. The distance between vertices u and v is denoted by d(u, v), is the length of a shortest path connecting u and v. The average distance between all pairs of vertices in G is denoted by l(g). That is l(g) = ( n ) u,v V (G) d(u, v), where the summation run over all unordered pairs of vertices. The eccentricity e G (v) of a vertex v in G is the largest distance from v to another vertex of G, i.e. max{d(v, w) w V (G)}. The diameter of G is the maximum eccentricity in G, This research was supported by NSFC (No. 606). Corresponding author. addresses: baoyinwu@gmail.com (Baoyindureng Wu), zwp99806@63.com (Wanping Zhang) Copyright c 0 DMFA Slovenije

2 Ars Math. Contemp. 7 (0) 5 denoted by diam(g). Similarly, the radius of G is the minimum eccentricity in G, denoted by rad(g); and the average eccentricity of G is denoted by ecc(g). In other words, rad(g) = min v V e G(v), diam(g) = max v V e G(v) and ecc(g) = e G (v). n For a connected graph G of order n, σ G (u) denotes the average distance from u to all other vertices of G, that is σ G (u) = n v V (G) d(u, v). The proximity π(g) of a connected graph G is the minimum average distance from a vertex of G to all others. Similarly, the remoteness ρ(g) of a connected graph G is the maximum average distance from a vertex of G to all others. They were recently introduced in [, 3], that is π(g) = min v V σ G(v) and ρ(g) = max v V σ G(v). The sum of distances from a vertex of G to all others is known as its transmission. Proximity and remoteness can also be seen as the minimum and maximum normalized transmission in a graph. Indeed, by their definitions v V π(g) rad(g) ecc(g) diam(g) and π(g) l(g) ρ(g) diam(g). There are a number of results which are devoted to the relation between average distance and other graph parameters (see [6-5, ]). A vertex u V (G) with the minimum eccentricity is called a center of G. It is well-known that every tree has either exactly one center or two, adjacent centers. The center of graphs have been extensively studied in the literature (see [6]). Some more results on the radius of graphs can be found in [8, 7]. A Soltés or a path-complete graph is the graph obtained from a clique and a path by adding at least one edge between an endpoint of the path and the clique. The Soltés graphs are known to maximize the average distance l when the number of vertices and of edges are fixed [0]. In [] Aouchiche and Hansen established the Nordhaus-Gaddum-type theorem for π(g) and ρ(g). In [5] the same authors gave the upper bounds on rad(g) π(g), diam(g) π(g) and ρ(g) π(g), and proposed five related conjectures, two of which are the following. Conjecture A. (Conjecture 5, [5]) Among all connected graphs G on n 3 vertices with average distance l and remoteness ρ, the Soltés graphs with diameter n+ maximize ρ l. Conjecture B. (Conjecture, [5]) Let G be a connected graph on n 3 vertices with remoteness ρ and radius r. Then connected graph G on n 3 vertices, ρ r { n n n, if n is even,, if n is odd. 3 n The inequality is best possible as shown by the cycle C n if n is even and of the graph composed by the cycle C n together with two crossed edges on four successive vertices of the cycle. The aim of this note is to confirm the validity of the above conjectures. Conjecture in [5] is solved in [9], and Conjecture in [5] is solved in []. Up to now, Conjecture 3 in [5] still remains open.

3 B. Wu and W. Zhang: Average distance, radius and remoteness of a graph 3 Proof of Conjecture A For convenience, we use some additional definition and notations. Let G be a connected graph. A vertex u V (G) is called a peripheral vertex if σ(u) = ρ(g). For a vertex u V (G), let V i (u) = {v V (G) d(u, v) = i} and n i (u) = V i (u) for each i {,..., d}, where d = e G (u). In what follows, V i (u) is simply denoted by V i for a peripheral vertex u of G. Lemma.. Let G be a connected graph of order n 3. Let u be a peripheral vertex of G and let d = e G (u). Let G be the graph obtained from G by joining each pair of all nonadjacent vertices x, y of G, where x, y V j V j+ for some j {,..., d }. We have ρ(g ) l(g ) ρ(g) l(g), with equality if and only if G = G. Proof. It is clear that for any x V (G), d G (u, x) = d G (u, x) and d G (v, w) d G (v, w) for any v, w V (G). It follows that σ G (u) = σ G (u) and σ G (v) σ G (v) for any v V (G ). Combining this with the assumption that u is a peripheral vertex of G, it follows that u is also a peripheral vertex of G. Thus ρ(g ) = σ G (u) = σ G (u) = ρ(g). Moreover, it is obvious that l(g ) l(g), with equality if and only if G = G. So, ρ(g ) l(g ) ρ(g) l(g), with equality if and only if G = G. Lemma.. Let G be a connected graph of order n 3. Let u be a peripheral vertex of G and e G (v) = d. Assume that G[V j V j+ ] is a clique for each j {0,..., d }. Let G be the graph with V (G ) = V (G) and E(G ) = E(G) {xy : x V d, y V d }. If d > n+, then ρ(g ) l(g ) ρ(g) l(g), with equality if and only if n is even and d = n +. Proof. Note that σ G (x) n n d, if x d j= V j {u} σ G (x) = σ G (x), if x V d σ G (x) n (n n d n d ), if x V d. Since u is a peripheral vertex of G, σ G (u) σ G (x) for any x d j= V j. Moreover, since d > n+, n d + n d n, with equality if and only if n is even and d = n +. Thus n n d n d n. Again by the assumption that u is a peripheral vertex of G, σ G (u) σ G (x) for any x V d. Also, for any y V d, σ G (y) = σ G (x) for any x V d. Thus σ G (u) > σ G (y). It means that u is a peripheral vertex of G, and ρ(g ) = σ G (u). So, ρ(g) ρ(g ) = σ G (u) σ G (u) = n n d. On the other hand, one can see that l(g) l(g ) = ( n ) [( ) ] [( ) ] n n d n d nd = n nd n d nd (n )n n n d. It follows that l(g) l(g ) ρ(g) ρ(g ), with equality if n is even and d = n +.

4 Ars Math. Contemp. 7 (0) 5 Lemma.3. Let G be a connected graph of order n 3. Let u be a peripheral vertex of G and e G (v) = d. Assume that G[V j V j+ ] is a clique for each j {0,..., d }. Let i be the smallest integer in {,..., d} such that n i (u). Let V i (u) = {u i } and v a vertex in V i (u). Denote by G the graph with V (G ) = V (G) and E(G ) = E(G) \ ({u i y : y V i \ {v}} A), where A = {vx : x V i+ } if i d, and A = otherwise. If d < n+, then ρ(g ) l(g ) > ρ(g) l(g). Proof. One can see that if i d, then n (n i ), if σ G (x) σ G (x) = n i, (i + ), if if i = d, then σ G (x) σ G (x) = n { n n d, (n d ), if x i j= V j {v, u} if x V i \ {v} x d j=i+ V j, x d j= V j {v, u} if x V d \ {v} If i d, then by d n+ n+, we have i + and n i > i +. Moreover, since u is a peripheral vertex of G, u is also a peripheral vertex of G. If i = d, then it is trivial to see that ) u is a peripheral vertex of G. So, ρ(g ) ρ(g) = σ G (u) σ G (u) = n ( n i. On the other hand, if i d, then l(g ) l(g) = ( n ) [( i + )( ) ( n i+ + n i+ + + n d + i ni )] = ( n ) [( i + )( ) ( n i n i + i ni )] = [( ) i + n i ] i n i. (n )n Define a function: f(i) = (n i ) n [(i + )n i i n i ]. By an easy calculation, one has f(i) = n 3(i + ) + n (i + i + n i ) and thus f (i) = 3 + n (i + ). Since i d, by d < n+, we have f (i) < 0. Thus f(i) is a decreasing function on [0, n+ n+ ], and achieves its minimum value at. One can check that f( n + ) > 0. Therefore f(i) > 0, and thus ρ(g ) ρ(g) > l(g ) l(g), the result follows. If i = d, then ρ(g ) ρ(g) = n ( n d ), and l(g ) l(g) = d(n d ) (n )n Since d n+, ρ(g ) ρ(g) l(g ) l(g) = n d >. d(n d ) =. (n )n

5 B. Wu and W. Zhang: Average distance, radius and remoteness of a graph 5 Lemma.. Let G be a connected graph of order n 3. Let u be a peripheral vertex of G and e G (v) = d. Assume that G[V j V j+ ] is a clique for each j {0,..., d } and that n i (u) for some i {,..., d }. Further, assume that i is the minimum subject to the above condition. Let v be a vertex in V i (u) and V i = {u i }. Let G be the graph with V (G ) = V (G) and E(G ) = E(G) A \ {vu i }, where A = {vy : y V i+ } if i d, and A = otherwise. If d = n+, then ρ(g ) l(g ) > ρ(g) l(g). Proof. Note that if i d, then n, if x i j= V j {u} σ G (x) σ G (x) = 0, if x V i V i+ if x d j=i+ V j, if i = d, then σ G (x) σ G (x) = n, { n, if x d j= V j {u} 0, if x V d V d. Thus ρ(g ) = σ G (u) = σ G (u) + n = ρ(g) + n. On the other hand, since i d = n < n, we have l(g ) l(g) = = < ( n ) [ i ( )] n i+ + n i n d ( ni + n i+ + i n ) (n )n (n )n i n. The results follows. The statement of Conjecture A is refined as follows. Theorem.5. Among all connected graphs G on n 3 vertices with average distance l and remoteness ρ, the maximum value of ρ l is attained by the Soltés graphs with diameter d, where { d = n+, if n is odd d { n, n + } if n is even. Proof. It is immediate from Lemmas.-.. In the remaining sections, we prove Conjecture B.

6 6 Ars Math. Contemp. 7 (0) 5 3 Some preparations Let G be a connected graph. Recall that for a vertex u V (G), let V i (u) = {v V (G) d(u, v) = i} and n i (u) = V i (u) for each i {,..., d}, where d = diam(g). Lemma 3.. Let G be connected graph with order n and radius r. If u is a center of G, then n i (u) for all i {,..., r }. Proof. By contradiction, suppose that n i (u) = for some i r and let V i (u) = {w}. Let P be a shortest path connecting u and w in G, v be the neighbor of u on P. For a vertex x V (G) \ {v}, { = d(u, x), if d(u, x) i d(v, x) d(u, x) +, if d(u, x) < i. It follows that ecc(v) = r, a contradiction. Corollary 3.. If G is a connected graph with order n and radius r, then r n. Proof. Let u be a center of G. By Lemma 3., n i (u) for all i {,..., r }. So, n + r i= n i(u) r, the result then follows. For a graph G, p(g) denotes the maximum cardinality of a subset of vertices that induce a path in G. Theorem 3.3. (Erdős, Saks, Sós [8]) For any connected graph G, p(g) rad(g). Corollary 3.. Let G be a connected graph of order n 3. For an even n, rad(g) = n if and only if G = P n or G = C n. Proof. The sufficiency is obvious. Next we prove its necessity. By Theorem 3.3, p(g) n. Let P = v... v n be an induced path of G, and let v n be the remaining vertex of G. We consider the vertex v n. Since d(v n, v i) < n for each i n, and rad(g) = n, we have d(v n, v n) = n. So, v nv i / E(G) for each i n, and thus N(v n ) {v, v n }. It implies that G {P n, C n }. For an odd integer n 5, we define some special graphs of order n with rad(g) = n : C n () is the graph obtained from C n by adding a new vertex which joins two adjacent vertices of C n ; C n () is the graph obtained from C n by adding a new vertex which joins two vertices with distance two on C n ; C n (3) is the graph obtained from C n by adding a new vertex which joins three consecutive vertices of C n. One can see that p(c n ) = p(c n ()) = n and p(c n )()) = p(c n (3)) = n. The construction of the following graphs are illustrated in Figure. For an i {,..., n }, P n (i, i, i+) is the graph obtained from P n by adding a new vertex which is adjacent to the vertices v i, v i, v i+ ; P n (i, i + ) is the graph obtained from P n by adding a new vertex which is adjacent to the vertices v i, v i+ ; P n (i, i + ) is the graph obtained from P n by adding a new vertex which is adjacent to the vertices v i, v i+ ; For j {,..., n }, P n (j) is the graph obtained from P n by adding a new vertex which adjacent to v j, where i, i + are taken modulo n.

7 B. Wu and W. Zhang: Average distance, radius and remoteness of a graph 7 Figure. Graphs with an odd order n, rad(g) = n and p(g) = n Note that P n (n, n, n+) = P n (n,, ) = C n () and P n (n, n) = C n. It is easy to see that p(p n (i, i, i+)) = p(p n (i, i+)) = p(p n (i, i+)) = n for each i {,..., n }, and p(p n (j)) = n for each j {,..., n }. The result of Lemma 3.5 is straightforward. But its proof is somewhat tedious and will be given in Section. Lemma 3.5. Let G be a connected graph of order n 5. If n is odd and rad(g) = n, then () p(g) = n if and only if G = P n () p(g) = n if and only if G {P n (i, i, i+), P n (i, i+), P n (i, i+ ) : i {,..., n }}, or G = P n (j) for some j {,..., n }. (3) p(g) = n if and only if G {C n (), C n (3)}. Corollary 3.6. Let G be a connected graph of order n 5. If n is odd and rad(g) = n then ρ(g) n+, with equality if and only if G {C n, C n (), C n (), C n (3)}. Proof. By Lemma 3.5, we consider the following cases. If G = P n, then ρ(g) = n i = n n > n +. i= Assume that either G = P n (, ) or G {P n (i, i, i + ), P n (i, i + ), P n (i, i + ), P n (i) : i {,..., n }. Let P = v... v n be the induced path of G, and v n be the new vertex, added to P in the construction of G. Since n 5, ρ(g) ρ(v ) > n i = n n i= n +. We saw that P n (n, ) = C n, P n (n, n, n + ) = C n () = P n (n,, ). It is easy to check that ρ(g) = n+ for G {C n, C n (), C n (), C n (3)} and ρ(p n (, n )) = ρ(p n (n, )) > n+.,

8 8 Ars Math. Contemp. 7 (0) 5 Now we are ready to prove Conjecture B. Theorem 3.7. Let G be a connected graph on n 3 vertices with remoteness ρ and radius r. Then with equality if and only if ρ r { n n n, if n is even,, if n is odd, 3 n { G = Cn, if n is even, G {C n, C n (), C n (), C n (3)}, if n is odd. Proof. If n = 3, then G = P 3 or G = K 3. Since ρ(p 3 ) = 3, ρ(k 3) =, and rad(p 3 ) = rad(k 3 ) =, ρ r 0, the result holds. Next we assume that n 5, and consider r ρ, instead of ρ r. Let u be a center of G, and n i = n i (u) for each i {,..., r}. Define a function f(r) = r n (n r + r ). By Corollary 3., since r n, f (r) = n (r ) > 0. Thus f(r) is a strictly increasing function on the interval [, n ], and achieves its maximum value n n n at r = n. Case. n is even By Lemma 3., n i for each i {,..., r }. Therefore, r ρ r n r in i i= r ( r ) (n r + ) + i + r n = r n (n r + r ) n n n. By Corollary 3., it is easy to check that r ρ = n Case. n is odd By the similar argument as in Case, we have i= r ρ r n (n r + r ) = f(r). n n if and only if G = C n. Since f(r) is a strictly increasing function on the interval [, n n ], if r, then

9 B. Wu and W. Zhang: Average distance, radius and remoteness of a graph 9 for n 5, f( n ) = ( n = n < n 3. ) (3 + (n ) ) n n 6 n So, it remains to consider the case when r = n. By Corollary 3.6, since ρ(g) n+, r ρ n n + n 3, with equality if and only if G {C n, C n (), C n (), C n (3)}. Proof of Lemma 3.5 () is trivial. The sufficiency of () is obvious by the construction of those graphs. To show the necessity of (), let P = v... v n be an induced path of G and v n be the remaining vertex of G. Claim. If v n has two neighbors v i, v j N(v n ) with i, j {,..., n }, then i j or i j n 3 = (n ). Proof of Claim. If n = 5, the cliam holds trivially. Next we show the claim by contradiction for n 7. Suppose that there exist two vertices v i, v j N(v n ) with i, j {,..., n } such that 3 i j n = (n ) 3. Without loss of generality, let i < j. Case. i n or j n+ By the symmetry, we just consider the case when i n. Note that d P (v n, v k ) < n for each k {,..., n }, d P (v n, v n ) = n, and d G (v n, v n ) d G (v n, v i ) +. Since 3 i j n = (n ) 3, we have d G (v n, v i ) n 3, and d G (v n, v n ) n. Furthermore d G (v n This proves that ecc(v n Case. i < n < n+ < j, v n ) d P (v n, v i ) + + d P (v j, v n ) < n. ) < n n, which contradicts rad(g) =.

10 50 Ars Math. Contemp. 7 (0) 5 We show that ecc(v n ) < n. Let C be the cycle obtained from the segment of P between v j and v j adding the vertex v n and joining it to v i and v j. It is clear that the length of C is at most n. So, for any vertex v on C, d(v n, v) C < n. To prove d(v n, w) < n, it suffices to show that max{d(v n, v ), d(v n, v n )} < n. This holds, because d G (v n, v ) d P (v n, v ) < n, d G (v n, v n ) d P (v n, v n ) < n. So, ecc(v n ) < n n, which contradicts rad(g) =. By Claim and p(g) = n, one has d(v n ) 3. Furthermore, if d(v n ) = 3, then N(v n ) = {v i, v i, v i+ } for some i {,..., n }, and thus G = P n (i, i, i+). Also, if d(v n ) =, then i j, and thus G {P n (i, i+), P n (i, i+)} for some i {,..., n }. If d(v n ) =, then by p(g) = n, G = P n (j) for some j {,..., n }. This completes the proof of (). The sufficiency of (3) is trivial. Next we show its necessity. By Theorem.3, let P = v... v n be an induced path of G, and v n, v n the remaining two vertices of G. Claim. Either N(v n ) \ {v n } = {v, v n } or N(v n ) \ {v n } = {v, v n }. Proof of Claim. By contradiction, suppose that Claim is not true. If there exist i, j {,..., n 3} such that v i N(v n ) and v j N(v n ), d(v n, v k ) n for k {n, n}. Together this with d(v n, v k ) n for k {,..., n }, we have ecc(v n ) n, a contradiction. Hence, either N(v n ) \ {v n } {v, v n } or N(v n ) \ {v n } {v, v n }. Without loss of generality, assume that N(v n ) \ {v n } {v, v n }. Since N(v n ) \ {v n } {v, v n } and p(g) = n, we have N(v n ) \ {v n } =. Moreover, since G is connected, we conclude that N(v n ) = {v n } and N(v n ) \ {v n, v, v n }. If there exists i {3,..., n } such that v i N(v n ) \ {v n, v, v n }, then it follows d(v n, v n ) n and thereby d(v n, v n ) n. So, ecc(v n ) < n, which contradicts rad(g) = n. This means that N(v n ) \ {v n, v, v n } {v, v n 3 }. Since N(v n ) \ {v n, v, v n }, let v N(v n ), without loss of generality. If n = 5, then by p(g) = 3, v, v 3 N(v 5 ), and thus e(v 5 ) =, a contradiction. For n 7, since p(g) = n, v n 3 N(v n ) or v n N(v n ). In both cases, one can see that ecc(v n ) max{ n 3, } < n. This proves Claim. By Claim, let N(v n ) \ {v n } = {v, v n }. Since P = v... v n is an induced path, G[{v,..., v n }] = C n. Claim 3. If v n has two neighbors v i, v j N(v n ) with i, j {,..., n }, then i j or i j n 3 = (n ).

11 B. Wu and W. Zhang: Average distance, radius and remoteness of a graph 5 Proof of Claim 3. By contradiction, suppose that v n has two neighbors v i, v j N(v n ) with i, j {,..., n } and 3 i j n. One can see that, for any vertex v k, d(v n, v k ) max{ i j +, n i j + } n 3 < n, it means that ecc(v n) < n, a contradiction. By Claim 3 and p(g) = n, one has d(v n ) 3. Furthermore, if d(v n ) = 3, then N(v n ) = {v i, v i, v i+ } for some i {,..., n }, and thus G = C n (3). Also, if d(v n ) =, then i j =, and thus G = C n (). This completes the proof of the necessity of (3). Acknowledgement The authors are grateful to the referees for their valuable comments. References [] X. An, B. Wu, Average distance and proximity of a graph, in preparation. [] M. Aouchiche, Comparaison Automatisée d Invariants en Théorie des Graphs, Ph.D. Thesis, École Polytechnique de Montréal, 006. [3] M. Aouchiche, G. Caporossi and P. Hansen, Variable neighborhood search for extremal graphs. 0. automated comparison of graph invariants, MATCH Commun. Math. Comput. Chem. 58 (007), [] M. Aouchiche and P. Hansen, Nordhaus-Gaddum relations for proximity and remoteness in graphs, Comput. Math. Appl. 59 (00), [5] M. Aouchiche and P. Hansen, Proximity and remoteness in graphs: results and conjectures, Networks 58 (0), [6] R. A. Beezer, J. E. Riegsecker, B. A. Smith, Using minimum degree to bound average distance, Discrete Math. 6 (00), [7] F. Buckley and F. Harary, Distance in Graphs, Addison-Wesley, Reading, MA, 990. [8] F. G. Chung, The average distance and the independence number, J. Graph Theory (988), [9] P. Dankelmann, Average distance and the independence number, Discrete Appl. Math. 5 (99), [0] P. Dankelmann, Average distance and the domination number, Discrete Appl. Math. 80 (997), 35. [] P. Dankelmann, Average distance and generalized packing in graphs, Discrete Math. 30 (00), [] P. Dankelmann and R. Entringer, Average distance, minimum degree, and spanning trees, J. Graph Theory 33 (000), 3. [3] P. Dankelmann, S. Mukwembi and H. C. Swart, Average distance and edge-connectivity I, SIAM J. Discrete Math. (008), 9 0. [] P. Dankelmann, S. Mukwembi and H. C. Swart, Average distance and edge-connectivity II, SIAM J. Discrete Math. (008), [5] E. DeLaVina and B. Waller, Spanning trees with many leaves and average distance, Elec. J. Combin. 5 (008), R33.

12 5 Ars Math. Contemp. 7 (0) 5 [6] R. C. Entringer, D. E. Jackson and D. A. Snyder, Distance in graphs, Czech. Math. J. 6 (976), [7] P. Erdős, J. Pach, R. Pollack and Z. Tuza, Radius, diameter, and minimum degree, J. Combin. Theory Ser. B 7 (989), [8] P. Erdős, M. Saks and V. T. Sós, Maximum induced trees in graphs, J. Combin. Theory Ser. B (986), [9] B. Ma, B. Wu and W. Zhang, Proximity and average eccentricity of a graph, Infor. Process. Lett. (0), [0] L. Soltés, Trnansmission in graphs: a bound and vertex removing, Math. Slovaca (99), 6. [] D. B. West, Introduction to Graph Theory, second Ed., Prentice Hall, Upper Saddle River, NJ, 00. [] B. Wu, G. Liu, X. An, G. Yan and X. Liu, A conjecture on average distance and diameter of a graph, Discrete Math. Alog. Appl. 3 (0),