# Bicyclic Graphs with the Second-Maximum and Third-Maximum Degree Resistance Distance

Authors: Wenjie Ning; Kun Wang; Hassan Raza
Journal: Journal of Mathematics
 (2021)

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## Abstract

LetG=V,E be a connected graph. The resistance distance between two vertices u and v in G, denoted by RGu,v, is the effective resistance between them if each edge of G is assumed to be a unit resistor. The degree resistance distance of G is defined as DRG=∑u,v⊆VGdGu+dGvRGu,v, where dGu is the degree of a vertex u in G and RGu,v is the resistance distance between u and v in G. A bicyclic graph is a connected graph G=V,E with E=V+1. This paper completely characterizes the graphs with the second-maximum and third-maximum degree resistance distance among all bicyclic graphs with n≥6 vertices.


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## Body

## 1. Introduction

All graphs considered in this paper are simple and undirected. LetG=V,E be a graph with n vertices and m edges. Let NGv be the set of vertices adjacent to v in G. The degree of v in G, denoted by dGv, is equal to NGv. Denote the minimum degree of vertices in G by δG. A vertex of degree one is called a pendant vertex, and the edge incident with a pendant vertex is called a pendant edge. The distance between two vertices u and v of G, denoted by dGu,v or du,v, is the length of a shortest path connecting u and v in G. For a subset S of V, denote by GS, the subgraph induced by S and G−S the graph GVG∖S. We use G−v instead of G−v if S=v for simplicity. Let Pn and Cn be the path and the cycle graphs on n vertices, respectively.A topological index or a graph-theoretic index is a real number related to a graph. Topological indices of molecular graphs are one of the oldest and most widely used descriptors in quantitative structure-activity relationships [1, 2]. One of the most exhaustively studied [3, 4] topological indices is the Wiener index. The Wiener index was introduced in 1947 [5] and defined as WG=∑u,v⊆VGdGu,v. It is well correlated with many physical and chemical properties of organic molecules and chemical compounds.Based on the electrical network theory, Klein and Randić [6] proposed a novel distance function called resistance distance in 1993. They treated a graph G as an electric network by considering each edge of G as a unit resistor. Then, the resistance distance between two vertices u and v in G, denoted by RGu,v, is defined as the effective resistance between them. Klein and Randić [6] also proved that RGu,v≤dGu,v, with equality if and only if there is a unique path connecting u and v in G. In recent years, this new type of distance between vertices in a graph has attracted prominent attention in mathematics and chemistry [6–11].Similar to the Wiener index, theKirchhoff index of a graph G is defined as(1)KfG=∑u,v⊆VGRGu,v.This invariant has wide applications in electric circuit, physical interpretations, chemistry, and graph theory [12–16].In 2012, Gutman et al. [17] introduced the concept of the degree resistance distance defined as(2)DRG=∑u,v⊆VGdGu+dGvRGu,v.Palacios called it asadditive degree-Kirchhoff index in [18]. In [17], Gutman et al. [17] presented some properties of DRG and characterized the unicyclic graphs with the minimum and second-minimum DRG. Later, the unicyclic graphs with the maximum and second-maximum DR-value were considered in [19, 20]. In [21, 22], the cactus graphs with the minimum, the second-minimum, and the third-minimum DR-values were also completely characterized. Recently, the bicyclic graphs with maximum and minimum DR-values were determined in [23, 24], respectively.Abicyclic graphG=V,E is a connected graph such that E=V+1. The kernel of G, denoted by G∧, is the unique bicyclic subgraph of G with no pendant vertices. Any bicyclic graph G is obtained from its kernel G∧ by attaching trees to some vertices in G∧. Given a family of graphs G, the graphs with the maximum and second maximum values of topological indices among G are examined widely, see in [25–29]. Motivated by this, in this paper, we determine the graphs with the second-maximum and third-maximum degree resistance distance among all bicyclic graphs with n≥6 vertices.


## 2. Preliminaries

Letℬn be the set of bicyclic graphs of order n, ℬn∞ be the set of bicyclic graphs of order n with exactly two cycles, and ℬnθ=ℬn∖ℬn∞. Let Bp,q be obtained from two vertex-disjoint cycles Cp and Cq by identifying a vertex u∈VCp and a vertex v∈VCq, Bp,l,q be obtained from two vertex-disjoint cycles Cp and Cq by connecting a vertex u∈VCp and a vertex v∈VCq by a path uv1v2…vl−1v of length ll≥1, and BPr,Ps,Pt be the union of three internally disjoint paths Pr, Ps, and Pt, respectively, with common end vertices, where r,s,t≥2 and at most one of them is 2.LetG be a graph and v be a vertex in G. Define KfvG=∑u∈VGRGu,v and DvG=∑u∈VGdGuRGu,v.We present a few lemmas which will be employed later to establish our main results.Lemma 1 (see [13]).
LetGbe a connected graph with a pendant vertexvwith its unique neighborw. Then, KfvG=KfwG−v+n−1.Lemma 2 (see [13]).
LetGbe a bicyclic graph of ordernandv∈VG. Then, KfvG≤n2/2−n/2−15/4. Moreover, ifdGv≥2, thenKfvG≤n2/2−3n/2+1/3.The following remark can be obtained from the proof of Lemma2.Remark 1.
LetGbe a graph inℬn∞andv∈VG. Then,KfvG≤n2/2−n/2−6.Lemma 3 (see [17]).
LetG be a connected graph with a cut vertex v such that G1 and G2 are two connected subgraphs of G having v as the only common vertex and VG1∪VG2=VG. Let n1=VG1,n2=VG2,m1=EG1,andm2=EG2. Then, DRG=DRG1+DRG2+2m2KfvG1+2m1KfvG2+n2−1DvG1+n1−1DvG2.Lemma 4 (see [17]).
LetCk be a cycle with length k and v∈Ck. Then, KfCk=k3−k/12,DRCk=k3−k/3, KfvCk=k2−1/6, and DvCk=k2−1/3.Lemma 5 (see [23]).
LetH be a connected graph of order h>2 and Ck be a cycle of order k≥4. Let F be the graph of order k obtained from C3 by attaching one pendant path of order k−3 to one vertex of C3. Further suppose G1 is the graph obtained from H and Ck by identifying one vertex in H and one vertex in Ck; G2 is the graph obtained from H and F by identifying one vertex in H and the pendant vertex in F. Then, we have DRG1<DRG2.By an argument similar to that of Lemma5, we easily get the following result.Lemma 6.
LetG be a connected graph of order n>2 and Ck be a cycle of order k≥5. Let F be obtained by identifying a pendant vertex of Pk−3 with any vertex of C4. Suppose G1is the graph obtained from G and Ckby identifying one vertex in G and one vertex in Ck;G2 is obtained from G and F by identifying one vertex in G and the pendant vertex in F. Then, DRG1<DRG2.In [23], Du and Tu characterized the unique bicyclic graph with maximum degree resistance distance. They also presented two significant lemmas in [23].Theorem 1 (see [23]).
LetGbe a bicyclic graph of ordern≥6; then,DRG≤2n3/3+n2−19n+88/3, with equality if and only ifG≅B3,n−5,3.Lemma 7 (see [23]).
LetGbe a bicyclic graph of ordernandv∈VG. Then,DvG≤n2+2n−73/4.Lemma 8 (see [23]).
LetGbe a bicyclic graph of ordern,vbe a pendant vertex ofG, andwbe its neighbor. Then,DRG=DRG−v+DwG−v+2KfwG−v+3n.


## 3. Bicyclic Graphs with the Second-Maximum Degree Resistance Distance

In this section, we will determine the bicyclic graphs with the second-maximum degree resistance distance.Supposen≥6. Let B3,n−5,3 be obtained from two 3-cycles v1v2v3v1 and vn−2vn−1vnvn−2 by connecting v3 and vn−2 by a path v3v4⋯vn−3vn−2. Define Gn1=B3,n−5,3−vn−1vn+vn−1vn−3 and Gn2,i=Gn1−vn−2vn+vivn, where 3≤i≤n−3. Let Gn3Gn5 be obtained from a 4-cycle C4=v1v2v3v4v1 and a path P=v5…vn by adding the edges v1v3 (v2v4, resp.) and v4v5. Let Gn4≅B4,n−6,3 be obtained from a 4-cycle v1v2v3v4v1 and a 3-cycle vn−2vn−1vnvn−2 by connecting v4 and vn−2 by a path v4v5…vn−3vn−2 (see Figure 1). Then, we have the following lemma.Figure 1
GraphsGn1,Gn2,i,Gn3,Gn4, and Gn5.Lemma 9.
LetGn1, Gn2,i, Gn3, Gn4, and Gn5 be defined as above. Then, DRGn1=2/3n3+n2−79/3n+56,DRGn2,i=2/3n3+n2−17n+4i2−4ni+88/3, DRGn3=2/3n3+n2−293/12n+117/2, DRGn4=2/3n3+n2−82/3n+167/3, and DRGn5=2/3n3+n2−163/6n+139/2.Proof.
By Lemma8 and Theorem 1, we easily obtain(3)DRGn1=DRGn1−vn+Dvn−2Gn1−vn+2Kfvn−2Gn1−vn+3n=23n−13+n−12−19n−1+883+2⋅23+3⋅23+2⋅23+1+2⋅23+2+⋯+2⋅23+n−7+3⋅23+n−6+2⋅43+n−6+2⋅43+n−6+2⋅23+23+23+1+23+2+⋯+23+n−6+243+n−6+3n=23n−13+n−12−19n−1+883+n2−143n+43+2⋅n22−176n+3+3n=23n3+n2−793n+56,DRGn2,i=DRGn2,i−vn+DviGn2,i−vn+2KfviGn2,i−vn+3n=23n−13+n−12−19n−1+883+2⋅1+2⋅2+⋯+2⋅i−4+3⋅i−3+4⋅i−3+23+2⋅1+2⋅2+⋯+2⋅n−4−i+3⋅n−3−i+4⋅n−3−i+23+2⋅1+2+⋯+i−3+2⋅i−3+23+1+2+⋯+n−3−i+2⋅n−3−i+23+3n=23n3−n2−19n+1463+n2+2i2−2ni−383+2⋅3n2−3n+6i2−6ni−206+3n=23n3+n2−17n+4i2−4ni+883.
LetH=Gn3v1,v2,v3,v4. By Lemma 3,(4)DRGn3=DRH+DRPn−3+2n−4Kfv4H+10Kfv4Pn−3+n−4Dv4H+3Dv4Pn−3=392+23n−33−n−32+13n−3+2⋅n−4⋅94+10⋅n−3n−42+n−4⋅234+3⋅n−42=23n3+n2−29312n+1172.
LetF=Gn4−vn−1,vn. By Lemmas 3 and 6,(5)DRGn4=DRC3+DRF+2n−2Kfvn−2C3+6Kfvn−2F+n−3Dvn−2C3+2Dvn−2F=8+23n−23−533n−2+48+83n−2+6n−222−n−22−72+83n−3+2n−22−11=23n3+n2−823n+1673.
LetS=Gn5v1,v2,v3,v4. By Lemma 3,(6)DRGn5=DRS+DRPn−3+2n−4Kfv4S+10Kfv4Pn−3+n−4Dv4S+3Dv4Pn−3=392+23n−43+n−42+13n−4+2n−4⋅74+10⋅n−3n−42+4n−4+3n−42=23n3+n2−1636n+1392.Theorem 2.
SupposeG is a graph in ℬn∞ with G≇B3,n−5,3 and n≥6. Then, DRG≤2/3n3+n2−79/3n+56, with equality if and only if G≅Gn1, where Gn1 is defined as in Lemma 9.Proof.
It is easy to verify that, for any graphG in ℬ6∞ with G≇B3,1,3, DRG≤78=2/3⋅63+62−79/3⋅6+56, with equality if and only if G≅G61.
Now, we assumen≥7 and consider the following two cases.Case 1δ(G) = 1: let v be a pendant vertex in G. If G−v≅B3,n−6,3, then either G≅Gn1, or G≅Gn2,i, where Gn1 and G≅Gn2,i are defined as in Lemma 9. By Lemma 9,(7)DRGn2,i=23n3+n2−17n+4i2−4ni+883≤23n3+n2−17n+4⋅32−4n⋅3+883<23n3+n2−793n+56.IfG−v≇B3,n−6,3, we prove it by induction on n. Let w be the neighbor of v. By the inductive hypothesis, Remark 1, and Lemmas 7–9(8)DRG=DRG−v+DwG−v+2KfwG−v+3n≤23n−13+n−12−793n−1+56+n−12+2n−1−734+2n−122−n−12−6+3n=23n3+n2−793n+64112<23n3+n2−793n+56.Case 2 (δG≥2): in this case, G is of the form Bp,q or Bp,l,q. By Lemmas 5 and 6, we have DRG≤DRGn4, with equality if and only if G≅Gn4. Note that DRGn4<DRGn1 by Lemma 9. Therefore, the proof is complete.Theorem 3.
SupposeGis a graph of ordern≥4inℬnθ. Then, DRG≤2/3n3+n2−293/12n+117/2, with equality if and only ifG≅Gn3, whereGn3is defined in Lemma 9.Proof.
It is easy to verify that the only graph inℬ4θ is G43 and DRG43=2/3⋅43+42−293/12⋅4+117/2. We assume n≥5 next, and consider the following two cases.Case 1 (δG=1): let v be a pendant vertex in G and w be the neighbor of v. We prove it by induction on n. By the inductive hypothesis, Lemma 2, and Lemmas 7–9,(9)DRG=DRG−v+DwG−v+2KfwG−v+3n≤23n−13+n−12−29312n−1+1172+n−12+2n−1−734+2⋅n−122−n−12−154+3n=23n3+n2−29312n+1172.The equalityDRG−v=2/3n3+n2−293/12n+117/2 holds if and only if DRG−v=2/3n−13+n−12−293/12n−1+117/2, DwG−v=n−12+2⋅n−1−73/4, and KfwG−v=n−12/2−n−1/2−15/4=n2/2−3/2n−11/4. By the inductive hypothesis, G−v≅Gn−13, which is obtained from a 4-cycle C4=v1v2v3v4v1 and a path P=v5…vn−1 by adding the edges v1v3 and v4v5. We show that w=vn−1, i.e., G≅Gn3.By direct calculation, we haveKfvn−1Gn−13=n2/2−3/2n−11/4, Kfv1Gn−13=Kfv3Gn−13=n2/2−31/8n+69/8<n2/2−3/2n−11/4, and Kfv2Gn−13=n2/2−7/2n+29/4<n2/2−3/2n−11/4. Obviously, KfuGn−13<Kfvn−1Gn−13 if u∈VGn−13∖v1,v2,v3,vn−1. Therefore, w=vn−1, i.e., G≅Gn3.Case 2 (δG≥2): then, G is of the form BPk,Pl,Pm. Suppose x and y are the only two vertices of degree 3. Since KfG≤1/8n3 (see [13]), we have(10)DRG=∑u,v⊆VGdu+dvRu,v=4KfG+KfxG+KfyG≤4⋅18n3+2⋅12n2−32n+13by Lemma2=12n3+n2−3n+23.Ifn≥10, then 1/2n3+n2−3n+2/3<2/3n3+n2−293/12n+117/2. For any graph G≅BPk,Pl,Pm when n=5,6,7,8,9, we have calculated DRG and found that DRG<2/3n3+n2−293/12n+117/2.
Combining Theorems1–3, we can obtain the first main result of our paper.Theorem 4.
SupposeGis a bicyclic graph of ordern≥6withG≇B3,n−5,3. Then,DRG≤2/3n3+n2−293/12n+117/2, with equality if and only ifG≅Gn3, whereGn3is defined as in Lemma 9.


## 4. Bicyclic Graphs with the Third-Maximum Degree Resistance Distance

In this section, we will determine the bicyclic graphs with the third-maximum degree resistance distance.Lemma 10.
LetGn3,ibe obtained from a 4-cycleC4=v1v2v3v4v1, a pathP=v5…vn−1and an isolated vertexvnby adding the edgesv1v3,v4v5, andvivn, where1≤i≤n−2andn≥6. Then,DRGn3,1=DRGn3,3=2/3n3+n2−455/12n+493/4,DRGn3,2=2/3n3+n2−437/12n+237/2,DRGn3,4=2/3n3+n2−485/12n+277/2, andDRGn3,i=2/3n3+n2−293/12n−4ni+4i2+4i+117/2, for5≤i≤n−2.Proof.
By Lemmas8 and 9, we easily obtain(11)DRGn3,1=DRGn3,3=DRGn−13+Dv1Gn−13+2Kfv1Gn−13+3n=23n−13+n−12−29312n−1+1172+2⋅58+3⋅58+3⋅12+2⋅58+1+⋯+2⋅58+n−6+n−5+58+2⋅58+58+12+58+1+⋯+58+n−5+3n=23n−13+n−12−29312n−1+1172+n2−354n+914+n2−314n+694+3n=23n3+n2−45512n+4934,DRGn3,2=DRGn−13+Dv2Gn−13+2Kfv2Gn−13+3n=23n−13+n−12−29312n−1+1172+3⋅58+3⋅58+3⋅1+2⋅2+⋯+2⋅n−5+n−4+2⋅58+58+1+⋯+n−4+3n=23n−13+n−12−29312n−1+1172+n2−8n+834+n2−7n+292+3=23n3+n2−43712n+2372,DRGn3,4=DRGn−13+Dv4Gn−13+2Kfv4Gn−13+3n=23n−13+n−12−29312n−1+1172+2⋅3⋅58+2⋅1+2⋅1+2⋅2+⋯+2⋅n−6+n−5+2⋅2⋅58+1+1+⋯+n−5+3n=23n−13+n−12−29312n−1+1172+n2−10n+1234+n2−9n+492+3n=23n3+n2−48512n+2772,and for 5≤i≤n−2,(12)DRGn3,i=DRGn−13+DviGn−13+2KfviGn−13+3n=23n−13+n−12−29312n−1+1172+2⋅1+2⋅2+⋯+2⋅n−2−i+n−1−i+2⋅1+2⋅2+⋯+2⋅i−5+3⋅i−4+2⋅3⋅i−4+58+2⋅i−3+2⋅1+2+⋯+n−1−i+1+2+⋯+i−3+2⋅i−4+58+3n=23n−13+n−12−29312n−1+1172+n2−2ni−2n+2i2+4i−694+n2−2ni−n+2i2−152+3n=23n3+n2−29312n−4ni+4i2+4i+1172.Proposition 1.
SupposeG≇Gn3is a bicyclic graph of ordern≥5andv∈VG, whereGn3is defined as in Lemma 9. Then,KfvG≤n2/2−n/2−17/4.Proof.
It is not hard to verify that, for any bicyclic graphG≇G53 of order 5 and v∈VG, KfvG≤52/2−5/2−17/4. Thus, we assume n≥6 in the following cases.Case 1 (dv=1): let w be the neighbor of v.SupposeG−v≅Gn−13, where Gn−13 is obtained from a 4-cycle C4=v1v2v3v4v1 and a path P=v5…vn−1 by adding the edges v1v3 and v4v5. Then, w≠vn−1 since G≇Gn3. By Lemma 1,(13)KfvG=KfwG−v+n−1≤maxKfvn−2Gn−13,Kfv2Gn−13+n−1=maxn22−52n+14,n22−72n+294+n−1=maxn22−32n−34,n22−52n+254<n22−n2−174.IfG−v≇Gn−13, we shall prove it by induction on n. By the inductive hypothesis, KfvG=KfwG−v+n−1≤n−12/2−n−1/2−17/4+n−1=n2/2−n/2−17/4.Case 2:dv≥2.
By Lemma2, KfvG≤n2/2−3n/2+1/3<n2/2−n/2−17/4.Lemma 11 (see [23]).
LetG be a bicyclic graph of order n, v be a pendant vertex of G, and w be its neighbor. Then, DvG=DwG−v+2n+1.Proposition 2.
LetG≇Gn3 be a graph in ℬnθ of order n≥5 and v∈VG, where Gn3 is defined as in Lemma 9. Then, DvG≤n2+2n−20.Proof.
It is easy to verify that for any graphG∈ℬ5θ with G≇G53 and v∈VG, DvG≤15=52+2⋅5−20. Thus, we assume n≥6 in the following cases.Case 1 (dGv=1): let w be the neighbor of v.SupposeG−v≅Gn−13, where Gn−13 is obtained from a 4-cycle C4=v1v2v3v4v1 and a path P=v5…vn−1 by adding the edges v1v3 and v4v5. Then, w≠vn−1 since G≇Gn3. Moreover, DvG=DwGn−13+2n+1 by Lemma 11. By direct calculation, we get Dv1Gn−13=Dv3Gn−13=n2−35/4n+91/4, Dv2Gn−13=n2−8n+83/4, Dv4Gn−13=n2−10n+123/4, and(14)DviGn−13=n2−2n+2i2+4−2ni−694≤n2−2n+2n−22+4−2nn−2−694=n2−2n−694=Dvn−2Gn−13,if5≤i≤n−2. Thus, DwGn−13≤Dvn−2Gn−13 and DvG≤Dvn−2Gn−13+2n+1=n2−65/4<n2+2n−20.IfG−v≇Gn−13, we prove it by induction on n. By the inductive hypothesis, DvG=DwG+2n+1≤n−12+2n−1−20+2n+1=n2+2n−20.Case 2:dGv≥2.Subcase 1:v is not contained by any cycle of G.By the same argument as that of Case 2 of Lemma 2.6 in [23], we can construct a series of bicyclic graphs G1,G2,…,Gk−1 in ℬnθ such that DvG<DvG1<⋯<DvGk−1 and v is a pendant vertex in Gk−1, where k=dGv≥2.SupposeGk−1≅Gn3. Then, Gk−1 is obtained from a 4-cycle C4=v1v2v3v4v1 and a path P=v5⋯vn−1v by adding the edges v1v3 and v4v5. By the transformation from Gk−2 to Gk−1, we can conclude that Gk−2=Gk−1−vn−2vn−1+vn−2v, i.e., Gk−2≅Gk−1. Note that DvGk−2=n2−73/4. We have DvG≤DvGk−2<n2+2n−20.IfGk−1≇Gn3, then, by Case 1, DvG<DvGk−1≤n2+2n−20.Subcase 2:v is in a cycle of G.
LetG∧ be the kernel of G. By Claims 1 and 2 of Lemma 2.6 in [23], we can construct a graph G″ in ℬnθ having G∧ as its kernel and DvG≤DvG″. Moreover, G″ is obtained from G∧ by attaching a pendant path to the vertex u, where u is a vertex of G∧ such that RGu,v=maxw∈VG∧RGw,v.
SupposeG″ has only two vertices of degree three, say w1 and w2. Without loss of generality, we assume that v≠w1, and v≠w2. Then, by Lemma 2,(15)DvG″=3RG″w1,v+RG″w2,v+∑w≠w1,w22RG″w,v=RG″w1,v+RG″w2,v+2KfvG″<dG″w1,v+dG″w2,v+2KfvG″≤n+2n22−32n+13<n2+2n−20.
SupposeG″ has exactly three vertices of degree three, say w1, w2, and w3. Let w4 be the pendant vertex of G″. Without loss of generality, we assume that v≠w1,w2,w3. Then, by Lemma 2,(16)DvG″=3RG″w1,v+RG″w2,v+RG″w3,v+RG″w4,v+∑w≠w1,w2,w3,w42RG″w,v<RG″w1,v+RG″w2,v+RG″w3,v+2KfvG″<dG″w1,v+dG″w2,v+dG″w3,v+2KfvG″≤3n−12+2⋅n22−32n+13≤n2+2n−20.
SupposeG″ has a vertex of degree four, say w1, and a vertex of degree three, say w2. Let w3 be the pendant vertex of G″. Without loss of generality, we assume that v≠w1,w2. Then, by Lemma 2,(17)DvG″=4RG″w1,v+3RG″w2,v+RG″w3,v+∑w≠w1,w2,w32RG″w,v<2RG″w1,v+RG″w2,v+2KfvG″<2dG″w1,v+dG″w2,v+2KfvG″≤3n−12+2⋅n22−32n+13≤n2+2n−20,which completes the proof.Theorem 5.
SupposeGis a graph of ordern≥5inℬnθ\Gn3. Then,DRG≤2/3n3+n2−163/6n+139/2, with equality if and only ifG≅Gn5, whereGn3andGn5are defined as in Lemma 9.Proof.
It is not hard to verify that, for any graphG in ℬ5θ\G53, DRG≤42=2/3⋅53+52−163/6⋅5+139/2, with equality if and only if G≅G55.
We assume thatn≥6, and consider the following two cases.Case 1:δG=1.Letvn be a pendant vertex of G. Suppose G−vn≅Gn−13, where Gn−13 is obtained from a 4-cycle C4=v1v2v3v4v1, and a path P=v5⋯vn−1 by adding the edges v1v3 and v4v5. Then, G≅Gn3,i, where 1≤i≤n−2, and Gn3,i is defined in the Lemmas 10. By Lemma 10,(18)DRGn3,1=DRGn3,3=23n3+n2−45512n+4934<23n3+n2−1636n+1392,DRGn3,2=23n3+n2−43712n+2372<23n3+n2−1636n+1392,DRGn3,4=23n3+n2−48512n+2772<23n3+n2−1636n+1392,DRGn3,i=23n3+n2−29312n−4ni+4i2+4i+1172≤23n3+n2−29312n−4nn−2+4n−22+4n−2+1172=23n3+n2−34112n+1332<23n3+n2−1636n+1392,for5≤i≤n−2.IfG−vn≇Gn−13, we prove it by induction on n. Let w be the neighbor of vn. By the inductive hypothesis, Lemma 8, and Propositions 1 and 2,(19)DRG=DRG−vn+DwG−vn+2KfwG−vn+3n≤23n−13+n−12−1636n−1+1392+n−12+2n−1−20+2⋅n−122−n−12−174+3n=23n3+n2−1636n+1392.The equalityDRG=2/3n3+n2−163/6n+139/2 holds if and only if DRG−vn=2/3n−13+n−12−163/6n−1+139/2, DwG−vn=n−12+2n−1−20, and KfwG−vn=n−12/2−n−1/2−17/4=n2/2−3/2n−13/4. By the inductive hypothesis, G−vn≅Gn−15, where Gn−15 is obtained from a 4-cycle C4=v1v2v3v4v1 and a path P=v5…vn−1 by adding the edges v2v4 and v4v5. We show that w=vn−1, i.e., G≅Gn5.By direct calculation, we haveKfvn−1Gn−15=n2/2−3/2n−13/4, Kfv2Gn−15=n2/2−4n+37/4<n2/2−3/2n−13/4, and Kfv1Gn−15=Kfv3Gn−15=n2/2−31/8n+73/8<n2/2−3/2n−13/4. Obviously, KfvGn−15<Kfvn−1Gn−15 if v∈VGn−15∖v1,v2,v3,vn−1. Therefore, w=vn−1, i.e., G≅Gn5.Case 2:δG≥2.
By a similar argument to that of Case 2 in Theorem3, we obtain(20)DRG≤12n3+n2−3n+23.
Ifn≥11, then 1/2n3+n2−3n+2/3<2/3n3+n2−163/6n+139/2. For any graph of the form BPk,Pl,Pm when n=6,7,8,9,10, we have calculated DRG and found that DRG≤2/3n3+n2−163/6n+139/2.
From Theorems2 and 4, we obtain the following result.Theorem 6.
LetGn1andGn5be defined as in Lemma 9. Then, among all bicyclic graphs of order n,(i)
If6≤n≤16, the graph Gn5 is the unique graph with the third-maximum degree resistance distance of value 2/3n3+n2−163/6n+139/2(ii)
Ifn≥17, the graph Gn1 is the unique graph with the third-maximum degree resistance distance of value 2/3n3+n2−79/3n+56


## 5. Conclusion

As a molecular structure descriptor, the Wiener index is one of the widely employed topological indices, as it is well correlated with many physical and chemical properties of a variety of classes of chemical compounds. A weighted version of the Wiener index is the degree resistance distance. In this paper, we characterize the graphs with the second-maximum and third-maximum degree resistance distance among all bicyclic graphs with fixed order. Furthermore, we present an open problem.Problem 1.
Characterize the tricyclic graphs of ordern with the maximum and second-maximum degree resistance distance.



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