Resistance Distances and Kirchhoff Indices Under Graph Operations

The resistance distance between any two vertices of a connected graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is defined as the net effective resistance between them in the electrical network constructed from <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> by replacing each edge with a unit resistor. The Kirchhoff index of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is defined as the sum of resistance distances between all pairs of vertices. In this paper, two unary graph operations on <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> are taken into consideration, with the resulted graphs being denoted by <inline-formula> <tex-math notation="LaTeX">$RT(G)$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$H(G)$ </tex-math></inline-formula>. Using electrical network approach and combinatorial approach, we derive explicit formulae for resistance distances and Kirchhoff indices of <inline-formula> <tex-math notation="LaTeX">$RT(G)$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$H(G)$ </tex-math></inline-formula>. It turns out that resistance distances and Kirchhoff indices of <inline-formula> <tex-math notation="LaTeX">$RT(G)$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$H(G)$ </tex-math></inline-formula> could be expressed in terms of resistance distances and graph invariants of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>. Our result generalizes the previously known result on the Kirchhoff index of <inline-formula> <tex-math notation="LaTeX">$RT(G)$ </tex-math></inline-formula> for a regular graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> to the Kirchhoff index of <inline-formula> <tex-math notation="LaTeX">$RT(G)$ </tex-math></inline-formula> for an arbitrary graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>.


I. INTRODUCTION
Let G be a connected graph with vertex set V (G) and edge set E(G). Suppose that V (G) = {v 1 , v 2 , . . . , v n }. It is well known that distance functions are of fundamental to a graph. The most natural and best known distance function defined on a graph is the (shortest-path) distance, where the distance between any two vertices of G is defined as the length of a shortest path connecting them. In 1993, a new novel distance function, resistance distance, was identified by Klein and Randić [1]. The concept of resistance distance originates from electrical circuit theory. If we view G as an electrical network N by replacing each edge of G with a unit resistor, then the resistance distance [1] between v i and v j , denoted by G (v i , v j ), is defined as the net effective resistance between the corresponding nodes in the electrical network N . In contrast to the shortest path distance, the resistance distance has a notable feature that if v i and v j are connected by more than one paths, then they are closer than they are connected by the only shortest path.
Beside being a distance function on graphs and an important component of electrical circuit theory, resistance distance The associate editor coordinating the review of this manuscript and approving it for publication was Tu Ngoc Nguyen .
has been found to have significant applications in chemistry. In comparison with shortest-path distance, resistance distance is more suitable to describe the fluid or wave-like communications in molecules. In particular, resistance distance-based graph invariants, turn out to play important roles in the study of QSAR (quantitative structure-activity relationship) and QSPR (quantitative structure-property relationship). The most widely used resistance distance-based graph invariant is the Kirchhoff index [1], which is defined as the sum of resistance distances between all pairs of vertices. In other words, the Kirchhoff index Kf (G) of G is defined as: Later, two modifications of the Kirchhoff index were introduced, which take the degrees of graphs into consideration. One is the multiplicative degree-Kirchhoff index, which is defined by Chen and Zhang [2] as: where d i is the degree (i.e., the number of neighbors) of the vertex v i . The other one is the additive degree-Kirchhoff index defined by Gutman et al. [3] as: In recent years, the computation of resistance distances and Kirchhoff indices of graphs under unary or binary operations has attracted much attention. In [4], Xu computed the Kirchhoff index of product and lexicographic product of two graphs. In [5], Zhang et al. derived explicit formulae for Kirchhoff indices of join, corona and cluster of two graphs. Then, Arauz [6] obtained the Kirchhoff index for generalized corona and cluster of networks. In [7], bounds for the degree Kirchhoff index of the line and para-line graphs were determined. Later, the Kirchhoff index in a composition of a rooted tree T and a graph G were studied in [8]. In [9], Yang and Klein obtained resistance distances in various composite graphs, such as join, product, composition, direct product, strong product, corona and rooted product. In [10], Bu et al. investigated resistance distance in subdivision-vertex join and subdivision-edge join of graphs. Then, Chen [11] obtained resistance distances and Kirchhoff indices of generalized join of graphs. Liu et al. [12] gave resistance distances and Kirchhoff indices of R-vertex join and R-edge join of two graphs. In [13], resistance distances for subdivision-vertex and subdivision-edge coronae were obtained. Then Kirchhoff indices of subdivision-vertex and subdivision-edge neighbourhood corona were obtained in [14] and [15]. In [16], Kirchhoff indices of n-prism networks (i.e. the graph obtained by the product of the path graph and the cycle graph) were obtained. After that, resistance distances and Kirchhoff indices were obtained for various kinds of corona of graphs, such as corona-vertex and the corona-edge of the subdivision graph [17], generalized subdivision-vertex and subdivision-edge corona [18], ordinary corona and neighborhood corona [19], generalized R-vertex and R-edge corona in [20]. After that, resistance distances and Kirchhoff indices of Q-double join graphs [21] and other two novel graph operations were obtained [22]. Besides these binary operations, special attention has been paid to resistance distances and related topological indices under unary operations, such as subdivision, triangulation, vertex-face operation, and so on. In [23], resistance distances in subdivision of a graph were determined. In [24], Gao et al. obtained the formula for the Kirchhoff index of subdivision of a regular graph. Then their result was generalized to the subdivision of general graphs in [25]. In [26], Wang et al. determined the Kirchhoff indices for triangulation T (G) (denoted by R(G) in their paper) and Q(G) of a regular graph G. Then Yang and Klein [27], Huang et al. [28] independently generalized their result to the Kirchhoff index of T (G) of a general graph G. In [29], Liu et al. studied a new graph operation RT (G) and obtained Kirchhoff index of RT (G) for a regular graph G. Later, they also obtained the Hosoya index of RT (G) in [30]. In [31], Shangguan and Chen obtained resistance distances in the vertex-face graph of a planar graph. In [32],  resistance distances and Kirchhoff indices of stellated graphs were obtained.
Motivated by these results, in this paper, we take two unary graph operations into consideration. For a graph G, let RT (G) and H (G) (detailed definitions will be given in the later) be the resulted graphs obtained from G by the two unary operations. First, resistance distances and the Kirchhoff index of RT (G) for a general graph G are determined, which generalized the result obtained by Liu et al. in [29]. Then, formulae for resistance distances and the Kirchhoff index of H (G) are derived. It turns out that resistance distances in RT (G) and H (G) could be expressed in terms of resistance distances of G, and Kirchhoff indices of RT (G) and H (G) could be expressed in terms of Kirchhoffian graph invariants (i.e. Kirchhoff index, multiplicative degree-Kirchhoff index, additive degree-Kirchhoff index) and parameters of G. Now we give detailed definitions of RT (G) and H (G). For a connected graph G, let T (G) be the triangulation of G, i.e.
Let RT (G) be the graph obtained from T (G) by adding a new edge corresponding to every vertex of G, and by joining each new edge to the corresponding vertex of G. For example, the graph G and corresponding RT (G) are given in Figure 1. Let be the maximum degree of G. Then the graph H (G) is defined to be the graph obtained from G by adding − d i pendent vertices to every vertex v i of G. For instance, the graph G and corresponding H (G) are depicted in Figure 2.

II. RESISTANCE DISTANCES AND KIRCHHOFF INDICES UNDER GRAPH OPERATIONS
In this section, we compute resistance distances and Kirchhoff indices of RT (G) and H (G) for an arbitrary graph G. For convenience, we divide this section into two subsections. VOLUME 8, 2020

A. RESISTANCE DISTANCES AND THE KIRCHHOFF INDEX OF RT (G)
Recall that V (G) = {v 1 , v 2 , . . . , v n }. We label the vertices of RT (G) in the following way: for each edge v k v l ∈ E(G), we label the vertex in RT (G) that associated with v k v l by v kl ; for each vertex v i ∈ V (G), we label the end-vertices of the edge in RT (G) that associate with v i (the edge newly To obtain resistance distances in RT (G), we first introduce the cut-vertex property in electrical circuit theory. For a con- Lemma 1: (The Cut-Vertex Property): Let G be a connected graph with v k being a cut vertex of G. If v i and v j are vertices which belong to different components in G−v k , then By the structure of RT (G), it is obvious that for any two vertices in V (T (G)), the resistance distance between them in T (G) is the same as that in RT (G). Thus to obtain resistance distances in RT (G), it is needed to introduce resistance distances in T (G), as given in the following Lemma. For simplicity, we denote the resistance distance functions of G, T (G), and RT (G) by , T and R , respectively. Lemma 2 ( [23], [27]): Let G be a connected graph. Then resistance distances in T (G) can be computed as follows: 2 On the basis of resistance distances in T (G), resistance distances in RT (G) could be given in the following result.
Theorem 1: Let G be a connected graph. Then resistance distances in RT (G) can be computed as follows: 3 Proof: Clearly, for any two vertices x, y belonging to V (T (G)) = V (G) ∪ V , T (x, y) = R (x, y). Thus Eqs. (6), (7) and (8) follows directly from Eqs. (3), (4) and (5) in Lemma 2. For any v i ∈ V (G), it is easily seen that Then, by the cut-vertex property, we have Thus Eq. (9) is derived by substituting the result in Eq. (6) into the above equation. In the same way, Eq. (10) could be derived from Eq. (7), and Eq. (11) could be derived from Eq. (6). In the following, according to Theorem 1, we compute the Kirchhoff index of RT (G).
Theorem 2: Let G be a connected graph with n ≥ 2 vertices and m edges. Then Proof: Since V (RT (G)) = V (T (G)) ∪ V , and any two vertices in T (G) has the same resistance distance as in RT (G). Hence For the Kirchhoff index of T (G), it has been shown in [27] that We proceed to compute the second term in the summation of Eq. (13). Since it is understood that Thus it follows by Theorem 1 that For the third term in the summation of Eq. (13), we divide it into two terms: For one thing, note that V (G) = {v 1 , v 2 , . . . , v n }. Then by Theorem 1, we have For another, by Theorem 1, we get u∈V v∈V If we let (v i ) be the sum of resistance distances between v i and all the other vertices of G, then the second terms in the summation of Eq. (18) becomes In addition, by the famous Foster's formula [33], which states that the sum of resistance distances between all pairs of adjacent vertices in a connected graph of order n is equal to Substituting Eqs. (20) and (19) back into Eq. (18), we have Then substituting Eqs. (17) and (21) back into Eq. (16), we get Finally, substituting Eqs. (14), (15) and (22) back into Eq. (13), we get as required.
In particular, if G is r-regular, then Kf + (G) = 2rKf (G), Kf * (G) = r 2 Kf (G), and m = rn 2 . Thus, as a straightforward consequence of Theorem 2, if G is a regular graph, then the Kirchhoff index of RT (G) could be expressed in a much simpler way, which is expressed only in terms of the Kirchhoff index, the number of vertices and the regularity degree of G.

B. RESISTANCE DISTANCES AND THE KIRCHHOFF INDEX OF H(G)
As before, suppose that V (G) = {v 1 , v 2 , . . . , v n }. Let d i be the degree of v i and let be the maximum degree of G. For For the sake of simplicity, we use (u, v) and H (u, v) to denote the resistance distance between vertices u and v in G and H (G), respectively. According to the structure of H (G) and the cut-vertex property, it is straightforward to obtain resistance distances in H (G), which are expressed in terms of resistance distances of G as follows.
Theorem 3: Let G be a connected graph. Then resistance distances in H (G) are given as follows. otherwise, By Theorem 3, we are able to give exact expression for the Kirchhoff index of H (G). It turns out that the Kirchhoff index of H (G) could be expressed in a quite neat way, only involves the Kirchhoffian indices, the maximum degree, the number of vertices and the number of edges of G.
Theorem 4: Let G be a connected graph with n vertices, m edges, and maximum degree . Then the Kirchhoff index of H (G) can be computed as follows. Proof: By Theorem 3, we have For the second term in the summation of Eq. (30), we have For the third term in the summation of Eq. (30), noticing For the fourth term in the summation of Eq. (30), we have Substituting results in Eqs.

III. CONCLUSION
In this paper, resistance distances and Kirchhoff indices under two kinds of unary graph operations are determined. It turns out that resistance distances and Kirchhoff indices of graphs under these operations could be expressed in terms of resistance distances and graph invariants of the original graph in a neat way. These formulae not only establish nice relations between resistance distances and Kirchhoff indices of the resulted complex graph and those of the original graph, but also make the computation of resistance distances and Kirchhoff indices of these complex graph become greatly simplified. Along this line, further study on resistance distances and Kirchhoff indices for some other unary graphs operations, such as total graph, medial graph, Mycielskian graph, is greatly anticipated.