All graphs considered in this article are simple and undirected. The resistance distance between vertices u and v of G was defined by Klein and Randi c ́ [1] to be the effective resistance between nodes u and v as computed with Ohm’s law when all the edges of G are considered to be unit resistors. The Kirchhoff index Kf(G) was defined in Ref. 1 as Kf(G) = ∑ _u<v r _uv , where r _uv (G) denotes the resistance distance between u and v in G. Resistance distance are, in fact, intrinsic to the graph, with some nice purely mathematical interpretations and other interpretations. The Kirchhoff index was introduced in chemistry as a better alternative to other parameters used for discriminating different molecules with similar shapes and structures [1]. The resistance distance and Kirchhoff index have attracted extensive attention due to their wide applications in physics, chemistry, and other fields. Until now, many results on the resistance distance and Kirchhoff index are obtained. The references in [2–5] can be referred to know more. However, the resistance distance and Kirchhoff index of the graph is, in general, a difficult thing from the computational point of view. The bigger the graph, the more difficult it is to compute the resistance distance and Kirchhoff index; so a common strategy is to consider a complex graph as a composite graph and to find relations between the resistance distance and Kirchhoff index of the original graphs. Let G = (V(G), E(G)) be a graph with the vertex set V(G) and edge set E(G). Let d _i be the degree of vertex i in G and D _G = diag (d ₁, d ₂, ⋯d _|V(G)|) the diagonal matrix with all vertex degrees of G as its diagonal entries. For graph G, let A _G and B _G denote the adjacency matrix and vertex-edge incidence matrix of G, respectively. The matrix L _G = D _G − A _G is called the Laplacian matrix of G, where D _G is the diagonal matrix of vertex degrees of G. We use μ ₁(G) ≥ u ₂(G) ≥⋯ ≥ μ _n (G) = 0 to denote the spectrum of L _G . For other undefined notations and terminology from graph theory, the readers may refer to Ref. 6 and the references therein [7–23]. The computation of the resistance distance between two nodes in a resistor network is a classical problem in electric theory and graph theory. For certain families of graphs, it is possible to identify a graph by looking at the resistance distance and Kirchhoff index. More generally, this is not possible. In some cases, the resistance distance and Kirchhoff index of a relatively larger graph can be described in terms of the resistance distance and Kirchhoff index of some smaller (and simpler) graphs using some simple graph operations. There are results that discuss the resistance distance and Kirchhoff index of graphs obtained using some operations on graphs, such as join, graph products, corona, and many variants of corona, such as edge corona and neighborhood corona. For such operations, it is possible to describe the resistance distance and Kirchhoff index of the resulting graph using the resistance distance and Kirchhoff index of the corresponding constituting graph; Refs. 14 and 15 can be referred for reference. This article considers the resistance distance and Kirchhoff index of the graph operations as follows, obtained from Ref. 11.

Definition 1

[11]: Let F, H _v be connected graphs, v be a specified vertex of H _v and u ₁, u ₂, … , u _k ∈ F. Let G = G[F, u ₁, u ₂, … , u _k , H _v ] be the graph obtained by taking one copy of F and k copies of H _v and then attaching the ith copy of H _v to the vertex u _i , i = 1, 2, … , k, at the vertex v of H _v (identify u _i with the vertex v of the ith copy). Then, the copies of the graph H _v that are attached to the vertices u _i , i = 1, 2, … , k are referred to as pockets, and G is described as a graph with k pockets.

Barik [11] has described the Laplacian spectrum of G = G [F, u ₁, u ₂, … , u _k , H _v ] using the Laplacian spectrum of F and H _v in a particular case when deg(v) = m. Recently, Barik and Sahoo [12] have described the Laplacian spectrum of more such graphs’ relaxing condition deg(v) = m. Let deg(v) = l, 1 ≤ l ≤ m. In this case, we denoted G = G [F, u ₁, u ₂, … , u _k , H _v ] more precisely by G = G [F, u ₁, u ₂, … , u _k ; H _v , l]. When k = n, we denoted simply by G = G [F; H _v , l]. If deg(v) = l, 1 ≤ l ≤ m, let N(v) = {v ₁, v ₂, … , v _l } ⊂ V(H _v ) be the neighborhood set of v in H _v . Let H ₁ be the subgraph of H _v induced by the vertices in N(v) and H ₂ be the subgraph of H _v induced by the vertices which are in V(H _v )\(N(v) ∪ {v}). When H _v = H ₁ ∨ (H ₂ + {v}), we described the resistance distance and Kirchhoff index of G = G[F, u ₁, u ₂, … , u _k , H _v ]. The graphs F = C ₄ and H − v = C ₃ are considered. Taking l = 1, 2 and 3, we obtained graphs G ₁ = G ₁ [F; H _v , 1], G ₂ = G ₂ [F; H _v , 2], and G ₃ = G ₃ [F; H _v , 3], respectively. Figure 1 is referred. In this case, we described the resistance distance and Kirchhoff index of G = G [F; H _v , l] in terms of the resistance distance and Kirchhoff index of F and H _v . The results are contained in Section 3 of this article. Furthermore, when F = F ₁ ∨ F ₂, F ₁ is the subgraph of F induced by the vertices u ₁, u ₂, … , u _k and F ₂ is the subgraph of F induced by the vertices u _k+1, u _k+2, … , u _n . The considered three graphs G ₂, G ₃, and G ₄ are shown in Figure 2, obtained from the two graphs F = K ₄ and H _v such that H _v \{v} = K ₃. It is observed that F = K ₁ ∨ K ₃, G ₂, G ₃, and G ₄ are graphs with 2, 3, and 4 pockets, respectively. Figure 2 can be referred. In this case, we described the resistance distance and Kirchhoff index of G[F, u ₁, u ₂, … , u _k ; H _v , l] in terms of the resistance distance and Kirchhoff index of F and H _v . These results are contained in Section 4.

The {1}-inverse of M is a matrix X such that MXM = M. If M is singular, then it has infinite {1}-inverse [16]. For a square matrix M, the group inverse of M, denoted by M ^# , is the unique matrix X such that MXM = M, XMX = X, and MX = XM. It is known that M ^# exists if and only if rank(M) = rank(M ²) [16, 17]. If M is really symmetric, then M ^# exists, and M ^# is a symmetric {1}-inverse of M. Actually, M ^# is equal to the Moore–Penrose inverse of M since M is symmetric [17].

It is known that the resistance distance in a connected graph G can be obtained from any {1}- inverse of G [13]. We used M ⁽¹⁾ to denote any {1}-inverse of a matrix M, and (M) _uv denotes the (u, v)-entry of M.

Lemma 2.1

[17]: Let G be a connected graph, then r u v G = L G 1 u u + L G 1 v v − L G 1 u v − L G 1 v u = L G # u u + L G # v v − 2 L G # u v . Let 1 _n denote the column vector of dimension n with all the entries equal to one. We often use 1 to denote all-ones column vector if the dimension can be read from the context.

Lemma 2.2

[14]: For any graph, we have L G # 1 = 0 .

Lemma 2.3

[18]: Let M = A B C D be a nonsingular matrix. If A and D are nonsingular, then M − 1 = A − 1 + A − 1 B S − 1 C A − 1 − A − 1 B S − 1 − S − 1 C A − 1 S − 1 = A − B D − 1 C − 1 − A − 1 B S − 1 − S − 1 C A − 1 S − 1 , where S = D − CA ⁻¹ B.

Lemma 2.4

[15]: Let L be the Laplacian matrix of a graph of order n. For any a > 0, we have L + a I n − a n J n × n # = L + a I − 1 − 1 a n J n × n .

Lemma 2.5

[5]: Let G be a connected graph on n vertices, then K f G = n t r L G 1 − 1 T L G 1 1 = n t r L G # .

Lemma 2.6

[19]: Let L = A B B T D be the Laplacian matrix of a connected graph. If D is nonsingular, then X = H # − H # B D − 1 − D − 1 B T H # D − 1 + D − 1 B T H # B D − 1 is a symmetric {1}-inverse of L, where H = A− BD ⁻¹ B ^T .

Let F be a connected graph with the vertex set {u ₁, u ₂, … , u _n }. Let H _v be a connected graph on m + 1 vertices with a specified vertex v and V(H _v ) = {v ₁, v ₂, … , v _m , v}. Let G = G[F; H _v , l]. It is noted that G has n(m + 1) vertices. Let deg(v) = l, 1 ≤ l ≤ m. With loss of generality, it is assumed that N(v) = {v ₁, v ₂, … , v _l }. Let H ₁ be the subgraph of H _v induced by the vertices in {v ₁, v ₂, … , v _l } and H ₂ be the subgraph of H _v induced by the vertices {v _l+1, v _l+2, … , v _m }. It is supposed that H _v = H ₁ ∨ (H ₂ + {v}). In this section, we focused on determining the resistance distance and Kirchhoff index of G[F; H _v , l] in terms of the resistance distance and Kirchhoff index of F, H ₁ and H ₂.

Theorem 3.1

Let G [F; H _v , l] be the graph, as described previously. It is supposed that H _v = H ₁ ∨ (H ₂ + {v}). Let the Laplacian spectrum of H ₁ and H ₂ be σ(H ₁) = (0 = μ ₁, μ ₂, … μ _l ) and σ(H ₂) = (0 = ν ₁, ν ₂, … ν _m−l ). Then, G [F; H _v , l] has the resistance distance and Kirchhoff index as follows:

(i) For any i, j ∈ V(F), we obtained

Proof: Let v j i denote the jth vertex of H in the ith copy of H _v in G, for i = 1, 2, … , n; j = 1, 2, … , m, and let V j ( H v ) = { v j 1 , v j 2 , … , v j n } . Then, V ( F ) ∪ ( ∪ j = 1 m V j ( H v ) ) is a partition of V(G). Using this partition, the Laplacian matrix of G = G[F; H _v , l] can be expressed as L G F ; H v , l = L F + l I n − 1 l T ⊗ I n 0 − 1 l ⊗ I n L H 1 + m − l + 1 I l ⊗ I n − J l × m − l ⊗ I n 0 − J m − l × l ⊗ I n L H 2 + l I m − l ⊗ I n . We began with the computation of {1}-inverse of the Laplacian matrix L(G) of G = G[F; H _v , l]. Let A = L(F) + lI _n , B = − 1 l T ⊗ I n 0 , B T = − 1 l ⊗ I n 0 and D = L H 1 + m − l + 1 I l ⊗ I n − J l × m − l ⊗ I n − J m − l × l ⊗ I n L H 2 + l I m − l ⊗ I n . First, we computed the D ⁻¹. By Lemma 2.3, we obtained A 1 − B 1 D 1 − 1 C 1 = L H 1 + m − l + 1 I l ⊗ I n − − J l × m − l ⊗ I n L H 2 + l I m − l − 1 ⊗ I n − J m − l × l ⊗ I n = L H 1 + m − l + 1 I l ⊗ I n − 1 l 1 m − l T L H 2 + l I m − l − 1 1 m − l 1 l T ⊗ I n = L H 1 + m − l + 1 I l ⊗ I n − m − l l J l × l ⊗ I n = L H 1 + m − l + 1 I l − m − l l J l × l ⊗ I n , so A 1 − B 1 D 1 − 1 C 1 − 1 = L H 1 + m − l + 1 I l − m − l l J l × l − 1 ⊗ I n . By Lemma 2.3, we obtained S − 1 = D 1 − C 1 A 1 − 1 B 1 − 1 = L H 2 + l I m − l ⊗ I n − − J m − l × l ⊗ I n L H 1 + m − l + 1 I l − 1 ⊗ I n − J l × m − l ⊗ I n − 1 = L H 2 + l I m − l ⊗ I n − J m − l × l L H 1 + m − l + 1 I l − 1 J l × m − l ⊗ I n − 1 = L H 2 + l I m − l − l m − l + 1 J m − l × m − l − 1 ⊗ I n . By Lemma 2.3, we obtained − A 1 − 1 B 1 S − 1 = − L H 1 + m − l + 1 I l − 1 ⊗ I n − J l × m − l ⊗ I n L H 2 + l I m − l − l m − l + 1 J m − l × m − l − 1 ⊗ I n = 1 m − l + 1 1 l ⊗ I n m − l + 1 l 1 m − l T ⊗ I n = 1 l J l × m − l ⊗ I n . Similarly, − S − 1 C 1 A 1 − 1 = ( − A 1 − 1 B 1 S − 1 ) T = 1 l J ( m − l ) × l ⊗ I n . So D − 1 = P 1 1 l J l × m − l ⊗ I n 1 l J m − l × l ⊗ I n Q 1 , where P 1 = [ L ( H 1 ) + ( m − l + 1 ) I l − m − l l J l × l ] − 1 ⊗ I n , Q 1 = [ ( L ( H 2 ) + l I m − l − l m − l + 1 J ( m − l ) × ( m − l ) ) ] − 1 ⊗ I n . Now, we computed the {1}-inverse of G[F; H _v , l]. By Lemma 2.6, we obtained H = A − B D − 1 B T = L F + l I n − − 1 l T ⊗ I n 0 P 1 1 l J l × m − l ⊗ I n 1 l J m − l × l ⊗ I n Q 1 − 1 l ⊗ I n 0 = L F + l I n − 1 l T ⊗ I n L H 1 + m − l + 1 I l − m − l l J l × l − 1 ⊗ I n 1 l ⊗ I n = L F + l I n − l I n = L F , so H ^# = L ^# (F). According to Lemma 2.6, we calculated − H ^# BD ⁻¹ and − D ⁻¹ B ^T H ^# . − H # B D − 1 = − L # F − 1 l T ⊗ I n 0 P 1 1 l J l × m − l ⊗ I n 1 l J m − l × l ⊗ I n Q 1 = L # F 1 l T ⊗ I n , L # F 1 m − l T ⊗ I n and − D − 1 B T H # = 1 l ⊗ I n L # F 1 m − l ⊗ I n L # F . We are ready to compute the D ⁻¹ B ^T H ^# BD ⁻¹. D − 1 B T H # B D − 1 = 1 l ⊗ I n L # F 1 m − l ⊗ I n L # F 1 l T ⊗ I n , 1 m − l T ⊗ I n = 1 l ⊗ I n L # F 1 l T ⊗ I n 1 l ⊗ I n L # F 1 m − l T ⊗ I n 1 m − l ⊗ I n L # F 1 l T ⊗ I n 1 m − l ⊗ I n L # F 1 m − l T ⊗ I n . Let P = [ L ( H 1 ) + ( m − l + 1 ) I l − m − l l J l × l ] ⊗ I n , Q = ( L ( H 2 ) + l I m − l − l m − l + 1 J ( m − l ) × ( m − l ) ) − 1 ⊗ I n , and M = 1 l J l × ( m − l ) ⊗ I n + ( 1 l ⊗ I n ) L # ( F ) ( 1 m − l T ⊗ I n ) ; then, based on Lemma 2.6, the following matrix N = L # F L # F 1 l T ⊗ I n L # F 1 m − l T ⊗ I n 1 l ⊗ I n L # F P 1 M 1 m − l ⊗ I n L # F M T Q 1 , (1) is a symmetric {1}-inverse of G[F; H _v , l], where P 1 = P − 1 + ( 1 l ⊗ I n ) L # ( F ) ( 1 l T ⊗ I n ) and Q 1 = Q − 1 + ( 1 m − l ⊗ I n ) L # ( F ) ( 1 m − l T ⊗ I n ) . For any i, j ∈ V(F), by Lemma 2.1 and Eq. 1, we obtained r i j G F ; H v , l = L # F i i + L # F j j − 2 L # F i j = r i j F , as stated in (i). For any i ∈ V(F) and j ∈ V(H ₁), by Lemma 2.1 and Eq. 1, we obtained r i j G F ; H v , l = L # F i i + L H 1 + m − l + 1 I l − m − l l J l × l − 1 ⊗ I n + 1 l ⊗ I n L # F 1 l T ⊗ I n j j − 2 L # F 1 l T ⊗ I n i j , as stated in (ii). For any i ∈ V(F) and j ∈ V(H ₂), by Lemma 2.1 and Eq. 1, we obtained r i j G F ; H v , l = L # F i i + L H 2 + l I m − l − l m − l + 1 J m − l × m − l − 1 ⊗ I n j j − 2 L # F i j , as stated in (iii). For any i ∈ V(H ₁) and j ∈ V(H ₂), by Lemma 2.1 and Eq. 1, we obtained r i j G F ; H v , l = L H 1 + m − l + 1 I l − m − l l J l × l − 1 ⊗ I n i i + L H 2 + l I m − l − l m − l + 1 J m − l × m − l − 1 ⊗ I n j j − 2 L H 1 + m − l + 1 I l − m − l l J l × l − 1 ⊗ I n i j , as stated in (iv). For any i ∈ V(H ₂) and j ∈ V(H ₁), by Lemma 2.1 and Eq. 1, we obtained r i j G F ; H v , l = L H 2 + l I m − l − l m − l + 1 J m − l × m − l − 1 ⊗ I n i i + L H 1 + m − l + 1 I l − m − l l J l × l − 1 ⊗ I n j j − 2 L H 2 + l I m − l − l m − l + 1 J m − l × m − l − 1 ⊗ I n i j , as stated in (v). Now, we computed the Kirchhoff index of G[F; H _v , l]. By Lemma 2.5, we obtained Kf(G[F; H _v , l]) = n m + 1 t r N − 1 T N 1 = n m + 1 t r L # F + t r L H 1 + m − l + 1 I l + m − l l J l × l − 1 ⊗ I n + + t r L H 2 + l I m − l + l m − l + 1 J m − l × m − l − 1 ⊗ I n + t r 1 l ⊗ I n L # F 1 l T ⊗ I n + t r 1 m − l ⊗ I n L # F 1 m − l T ⊗ I n − 1 T N 1 .

It is noted that the eigenvalues of (L(H ₂) + lI _m−l ) are 0 + l, ν ₂(H ₂) + l, … , ν _m−l (H ₂) + l and the eigenvalues of J _{(m−l)×(m−l)} are (m − l), 0^(m−l−1). Then, t r L H 2 + l I m − l − l m − l + 1 J m − l − 1 ⊗ I n = n ∑ i = 2 m − l 1 ν i H 2 + l + n m − l + 1 l . (2) Similarly, t r L H 1 + m − l + 1 I l − m − l l J l × l − 1 ⊗ I n = n ∑ i = 2 l 1 μ i H 1 + m − l + 1 + n . It is easily obtained t r 1 l ⊗ I n L # F 1 l T ⊗ I n + t r 1 m − l ⊗ I n L # F 1 m − l T ⊗ I n = t r J l × l ⊗ L # F + t r J m − l × m − l ⊗ L # F = l t r L # F + m − l t r L # F = m t r L # F . (3) Let P = L ( H 1 ) + ( m − l + 1 ) I l − m − l l J l × l ⊗ I n , then 1 T P − 1 1 = 1 l T 1 l T ⋯ 1 l T P − 1 0 0 … 0 0 P − 1 0 … 0 0 0 … … 0 0 0 0 … P − 1 1 l 1 l ⋯ 1 l , (4) = l 1 l T L H 1 + m − l + 1 I l − m − l l J l × l − 1 1 l = l 2 . Let Q = L ( H 2 ) + l I m − l − l m − l + 1 J ( m − l ) × ( m − l ) ⊗ I n , then 1 T Q − 1 1 = 1 m − l T 1 m − l T ⋯ 1 m − l T Q − 1 0 0 … 0 0 Q − 1 0 … 0 0 0 … … 0 0 0 0 … Q − 1 1 m − l 1 m − l ⋯ 1 m − l , (5) = m − l 1 m − l T L H 2 + l I m − l − l m − l + 1 J m − l × m − l − 1 1 m − l = m − l 2 m − l + 1 l . 1 ln T 1 l ⊗ I n L # F 1 l T ⊗ I n 1 ln = 1 n T 1 n T ⋯ 1 n T I n I n ⋯ I n , (6) L # F I n I n ⋯ I n 1 n 1 n ⋯ 1 n = n 2 1 n T L # F 1 n = 0 .

Similarly, 1 T ( 1 l ⊗ I n ) L # ( F ) ( 1 m − l T ⊗ I n ) 1 = 0 , 1 T ( 1 m − l ⊗ I n ) L # ( F ) ( 1 l T ⊗ I n ) 1 = 0 and 1 T ( 1 m − l ⊗ I n ) L # ( F ) ( 1 l T ⊗ I n ) 1 = 0

Plugging Eqs 2–6 and the aforementioned equations into Kf(G[F; H _v , l]), we obtained the required result in (vi).

In this section, we considered the case when F = F ₁ ∨ F ₂, where F ₁ is the subgraph of F induced by the vertices u ₁, u ₂, … , u _k and F ₂ is the subgraph of F induced by the vertices u _k+1, u _k+2, … , u _n . In this case, we indicated the explicit formulae of the resistance distance and Kirchhoff index of G = G[F, u ₁, u ₂, … , u _k ; H _v , l] in terms of the resistance distance and Kirchhoff index of G and H _v .