Bounds on the general eccentric distance sum of graphs

Some sharp bounds on the general eccentric distance sum are presented for (i) graphs with given order and chromatic number, (ii) trees with given bipartition, and (iii) bipartite graphs with given order and matching number. Bounds for bipartite graphs hold also if the matching number is replaced by the independence number, vertex cover number or edge cover number.


Introduction
Let V (G) and E(G) be the vertex set and edge set of a graph G. The number of vertices is called the order and the number of edges is the size of G. The number of edges incident with a vertex u is the degree deg G (u) of u. The number of edges in a shortest path between vertices u and v is the distance d G (u, v) between u and v. The distance between u and a vertex farthest from u in G is the eccentricity ecc G (u) of u in G. The diameter of G is the maximum eccentricity among eccentricities of the vertices in G.
A matching is a set of edges of a graph G such that no two edges in that set have a vertex in common. A vertex independent set is a set of vertices of a graph G such that no two vertices in that set are adjacent in G. The cardinality of a maximum matching/independent set is the matching number/independence number of G, respectively. A vertex cover of a graph G is a set of vertices such that each edge of G is incident with at least one vertex from that set. An edge cover of G is a set of edges such that each vertex of G is incident with at least one edge from that set. The vertex/edge cover number is the cardinality of a minimum vertex cover/edge cover, respectively.
For k ≥ 2, a graph is called k-partite if its vertex set can be partitioned into k sets, where any two vertices from the same set are non-adjacent. A complete k-partite graph K p1,p2,...,p k is a k-partite graph with partite sets of orders p 1 , p 2 , . . . , p k , where any two vertices from different partite sets are adjacent. A 2-partite graph is called a bipartite graph.
A connected graph containing no cycles is a tree. A leaf is a vertex of a tree having degree 1. The double star S p1,p2 is a tree containing exactly two vertices which are not leaves, and their degrees are p 1 and p 2 , respectively. So S p1,p2 contains p 1 + p 2 − 2 leaves. For the complement G of G, we have V (G) = V (G) and uv ∈ E(G) if and only if vertices u and v are not adjacent in G.
For a connected graph G and a, b ∈ R, the general eccentric distance sum is defined as . This index generalizes several distance-based indices. We obtain the classical eccentric distance sum for a = 1 and b = 1, the total eccentricity index for a = 1 and b = 0, and the first Zagreb eccentricity index of G for a = 2 and b = 0. For a = 0 and b = 1, we get EDS 0,1 (G) = 2W (G), where W (G) is the Wiener index. The eccentric distance sum EDS belongs to topological indices which have been investigated extensively. An upper bound on the EDS for graphs of given order and minimum degree was obtained by Mukungunugwa and Mukwembi [16]. A lower bound for trees with prescribed order was given by Yu, Feng, and Ilić [24], and also by Hua, Xu, and Wen [11]. The EDS for trees was studied also in [8,18]. The EDS was investigated for several basic graphs in [17], graphs related to groups in [1], cubic transitive graphs in [23], graph operations in [2], bipartite graphs in [4,15], fullerances in [9], and Sierpiński networks in [3]. Relations between the EDS and a few other indices were studied in [10]. Interesting results on the EDS were presented also in [12][13][14]. Bounds on EDS a,b for trees, bipartite graphs and general graphs with given order as well as for graphs with given order and number of pendant vertices/vertex connectivity were presented in [20]. Another general index was studied for example in [6] and some distance-based indices were investigated also in [5,7,19].
We give bounds on EDS a,b for trees with given bipartition and bipartite graphs with given order and matching number. Lower bounds are obtained for a ≥ 0 and 0 < b ≤ 1. Upper bounds are obtained for a ≤ 0 and b < 0. Bounds for bipartite graphs hold also if the matching number is replaced by the independence number, vertex cover number or edge cover number. A lower bound on EDS a,b for graphs with given order and chromatic number, where each color is used for at least two vertices, is presented for a ≥ 0 and b ≥ 1. Finally, for a ≥ 0, we present a lower bound on EDS a,1 (G) for graphs G of given order and size containing no vertex adjacent to all the other vertices, and a lower bound on EDS a,1 (G) + EDS a,1 (G) for graphs G of given order. All the bounds are sharp and extremal graphs are presented.

Results
First, we present Lemma 2.1 which was proved in [21]. We use this lemma in the proofs of Theorems 2.1, 2.2, 2.3, 2.4, and 2.5 to compare EDS a,b of some graphs.
A bipartite graph with two partite sets U 1 and U 2 has an (s, t)-bipartition if |U 1 | = s and |U 2 | = t. Clearly, for the order n of G, we have n = s + t. In Theorems 2.1 and 2.2, we consider trees having an (s, t)-bipartition with s ≥ t ≥ 2, because the unique tree having an (s, 1)-bipartition is the star with s + 1 vertices. For a = b = 1, Theorem 2.1 was presented in [8]. For a = 2 and b = 0, Theorem 2.1 was given in [19]. Theorem 2.1. Let a ≥ 0, 0 < b ≤ 1 and s ≥ t ≥ 2. For any tree T with an (s, t)-bipartition, with equality if and only if T ∼ = S s,t .
Proof. Among trees with an (s, t)-bipartition, we denote a tree with the smallest EDS a,b by T . Let us prove by contradiction that T ∼ = S s,t .
Assume that T S s,t . A tree with diameter d ≤ 2 does not exist for s ≥ t ≥ 2 and the only tree having diameter 3 is S s,t . Thus d ≥ 4. We denote a diametral path in T by u 0 u 1 . . . u d (so d T (u 0 , u d ) = d) and the leaves adjacent to u d−1 by w 1 , w 2 , . . . , w p , where u d is one of them and p ≥ 1. Without loss of generality, we assume that Clearly, T has an (s, t)-bipartition. Let us use u 1 and u d−1 to obtain a contradiction. We have ecc for a ≥ 0 and 0 < b < 1. Moreover, there are some vertices for those vertices. Thus, We have a contradiction. So T ∼ = S s,t which contains two vertices which are not leaves, For any tree T with an (s, t)-bipartition, with equality if and only if T ∼ = S s,t .
Proof. Only those parts of the proof are presented which differ from the proof of Theorem 2.1. Among trees with an (s, t)bipartition, we denote a tree with the largest EDS a,b by T . Since because for b < 0, from Lemma 2.1, we obtain Therefore EDS a,b (T ) < EDS a,b (T ). Hence T does not have the maximum EDS a,b . We have a contradiction.
The proofs of Theorems 2.3, 2.4 and 2.5 use the next lemma which was proved in [20]. For a ≤ 0 and b < 0, we have EDS a,b (G + uv) > EDS a,b (G).
Any graph has the matching number ν at most n 2 . Stars are the unique connected bipartite graphs with matching number 1. Thus, let us consider bipartite graphs for 2 ≤ ν ≤ n 2 . For a = b = 1, Theorem 2.3 was presented in [15].
with equality if and only if G ∼ = K ν,n−ν .
Proof. Among bipartite graphs of order n and matching number ν, let us denote a graph with the minimum EDS a,b by G . Without loss of generality, suppose that |U 1 | ≤ |U 2 |, where U 1 and U 2 are the partite sets of G . Let us show by contradiction that G ∼ = K ν,n−ν . Suppose that G K ν,n−ν . Clearly, |U 1 | ≥ ν (otherwise if |U 1 | ≤ ν − 1, the matching number of G is at most ν − 1). Note that G cannot be a subgraph of K ν,n−ν , (if G would be a subgraph of K ν,n−ν , by Lemma 2.2, we get EDS a,b (G ) > EDS a,b (K ν,n−ν )). Therefore ν < |U 1 | ≤ |U 2 |.
Let us denote any matching in G having ν edges by M . For j = 1, 2, let U ν j be the subset of U j containing ν vertices incident with the edges in M . We have |U j | = ν + l j with l j > 0 (and 2ν + l 1 + l 2 = n). Obviously, a vertex u 1 ∈ U 1 \ U ν 1 is not adjacent to a vertex u 2 ∈ U 2 \ U ν 2 , otherwise there would be the matching M ∪ {u 1 u 2 } in G with ν + 1 edges. Let us define the graph H having the same vertices as G , and containing all the edges between U ν 1 and U ν 2 , between U ν 1 and U 2 \ U ν 2 , and between U 1 \ U ν 1 and U ν 2 . The graph G is a subgraph of H , thus from Lemma 2.2, we obtain EDS a,b (H ) < EDS a,b (G ). The matching number of H is at least ν + 1.
Now we construct a graph H from H by deleting all the edges between U 1 \ U ν 1 and U ν 2 , and by adding all the edges between U 1 \ U ν 1 and U ν 1 . The graph H is bipartite, having the matching number ν and H is a subgraph of K ν,n−ν . Thus, by Lemma 2.2, we have EDS a,b (K ν,n−ν ) < EDS a,b (H ).
Proof. Only those parts of the proof are presented which differ from the proof of Theorem 2.3. Among bipartite graphs of order n and matching number ν, let us denote a graph with the maximum EDS a,b by G . The graph G is a subgraph of H , thus by Lemma 2.2, we get EDS a,b (H ) > EDS a,b (G ). Similarly, by Lemma 2.2, we have EDS a,b (K ν,n−ν ) > EDS a,b (H ). For j = 1, 2, we have D H (u j ) > D H (u j ) and ecc H (u j ) > ecc H (u j ), thus for a ≤ 0 and b < 0, we obtain Then Consequently, since by Lemma 2.1, Let us denote the independence number, vertex cover number and edge cover number by α, β and β , respectively. It is known that for a graph of order n, α + β = n; see [22]. If G has no isolated vertices, then ν + β = n.
Corollary 2.1. For a connected bipartite graph G of order n and vertex cover number β, where 2 ≤ β ≤ n 2 , we have if a ≤ 0 and b < 0. The equalities hold if and only if G ∼ = K β,n−β .
Since ν + β = n, Theorem 2.3 says that if a ≥ 0 and 0 < b ≤ 1, then for a connected bipartite graph G of order n and matching number n − β , we get Similarly, an upper bound can be obtained for a ≤ 0 and b < 0. Thus, we obtain Corollary 2.2.

Corollary 2.2.
For a connected bipartite graph G of order n and edge cover number/independence number β , where if a ≤ 0 and b < 0. The equalities hold if and only if G ∼ = K β ,n−β .
The smallest number of colors needed to color the vertices of a graph G such that no two adjacent vertices have the same color is the chromatic number of G. Clearly, any nonempty graph has chromatic number at least 2. Graphs with given order and chromatic number are investigated in Theorem 2.5. For a = b = 1, Theorem 2.5 was presented in [14]. Proof. For the considered set of graphs, let us denote a graph with the minimum EDS a,b by G . Since the graph G does not contain edges between vertices colored by the same color, G is a χ-partite graph. Each of the χ colors is used for at least two vertices, thus each partite set contains at least two vertices. By Lemma 2.2, any two vertices from different partite sets are adjacent, therefore G ∼ = K p1,p2,...,pχ , where p j ≥ 2, j = 1, 2, . . . , χ. Let us show that |p j − p l | ≤ 1 for any 1 ≤ j < l ≤ χ.
In Theorems 2.6 and 2.7, we obtain bounds on EDS a,b for b = 1. For a = 1, the bound given in Theorem 2.6 was presented in [13]. Proof. Since no vertex of G is adjacent to all the other vertices, we have ecc G (u) ≥ 2 for every u ∈ V (G). Thus, For any graph G, we have We show that there exist graphs with a small size (size close to n − 1) as well as some graphs with a large size (size close to n(n−1) 2 ) which attain the bound presented in Theorem 2.6. Note that C n is the cycle of order n and n 2 K 2 (for even n) is the set of n 2 independent edges. The graphs K c,n−c for 2 ≤ c ≤ n 2 , n 2 K 2 for even n and C n have diameter 2 and they do not contain a vertex of degree n − 1, therefore they belong to the extremal graphs for Theorem 2.6. The graphs K c,n−c have a small size if c is small. We have |E(K c,n−c )| = c(n − c). Thus, for c = 2, |E(K 2,n−2 )| = 2n − 4. The graphs n 2 K 2 for even n and C n have a large size. We have E n 2 K 2 = n(n − 2) 2 and E C n = n(n − 3) 2 .
For a = 1, the bound given in Theorem 2.7 was presented in [12,13].