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Bounds on F-index of tricyclic graphs with fixed pendant vertices

  • Sana Akram , Muhammad Javaid EMAIL logo and Muhammad Jamal
Published/Copyright: March 13, 2020
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Open Mathematics
From the journal Open Mathematics Volume 18 Issue 1

Abstract

The F-index F(G) of a graph G is obtained by the sum of cubes of the degrees of all the vertices in G. It is defined in the same paper of 1972 where the first and second Zagreb indices are introduced to study the structure-dependency of total π-electron energy. Recently, Furtula and Gutman [J. Math. Chem. 53 (2015), no. 4, 1184–1190] reinvestigated F-index and proved its various properties. A connected graph with order n and size m, such that m = n + 2, is called a tricyclic graph. In this paper, we characterize the extremal graphs and prove the ordering among the different subfamilies of graphs with respect to F-index in Ωnα , where Ωnα is a complete class of tricyclic graphs with three, four, six and seven cycles, such that each graph has α ≥ 1 pendant vertices and n ≥ 16 + α order. Mainly, we prove the bounds (lower and upper) of F(G), i.e

8n+12α+76≤F(G)≤8(n−1)−7α+(α+6)3 for each G∈Ωnα.
Keywords: extremal graphs; tricyclic graphs; F-index
MSC 2010: 05C12; 05C50; 05C35

1 Introduction and preliminaries

A representative number of a molecular graph that expresses the various features of the involved organic molecules, usually known as a topological index (TI). It plays an important role to study the certain changes in the molecular structures which may be physical or chemical. Moreover, Cheminformatics studies quantitative structural activity and property relationships that are used to examine the bioactivities and chemical reactivities of the chemical compounds in a molecular graph on the bases of obtained computational results for the different topological indices (TI’s), see [1]. Most importantly, all the TI’s are invariants under the parameter of graphs-isomorphism. For a connected graph, there are many TI’s in literature. These are classified into three main classes degree-based TI’s, distance-based TI’s and polynomial-based TI’s. The TI’s depending upon degrees are more familiar than the others, see [2].

Wiener (1947) defined the first distance based TI, when he was working on paraffin, see [3]. Later on, it was called by Wiener index and much more work has been done on it. Recently, Furtula and Gutman (2015) [4] reinvestigated a degree-based TI and named it forgotten index (F-index). They also proposed its basic properties in the same paper and reported that it can enhance the physico-chemical capability of the molecules. The F-index and its co-index of the different graphs are studied by De et al. [5], Milovanovic et al. [6] and Basavanagoud et al. [7]. Khaksari and Ghorbani [8] studied the certain product of graphs with the same index. The extremal graphs with respect to F-index among the unicyclic and bicyclic graphs are studied in [9, 10]. For more studies, we refer to [11] and [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25].

In this paper, we prove the existence of extremal graphs with respect to F-index in the class of tricyclic graphs with three, four, six and seven cycles under the condition of certain pendant vertices. We also investigate the ordering and compute the bounds (lower and upper) of the F-index in the same class of graphs.

Throughout the paper, G(V(G), E(G)) for vertex-set V(G) and edge-set E(G) is considered as simple (no loops and parallel edges), finite and undirected graph. For r ∈ V(G), d(r) shows its degree (number of incident edges on r). For more theoretic terminologies, we refer [26]. Now, some important TI’s are defined as follows:

Definition 1.1

For a (molecular) graph G, the first and second Zagreb indices are

M1(G)=∑rs∈E(G)[d(r)+d(s)]aandaM2(G)=∑rs∈E(G)[d(r)×d(s)].

Definition 1.2

For a (molecular) graph G, the general Randić index (Rα(G)) is

Rα(G)=∑rs∈E(Γ)[d(r)×d(s)]α.

For α = – 12 , α = 12 and α = 1, we obtain Randić, reciprocal Randić and second Zagreb indices respectively.

Definition 1.3

For a (molecular) graph G the forgotten index (F-index) is defined as follow:

F(G)=∑s∈V(G)[d(s)]3.

For more studies, we refer to [4, 11, 27, 28, 29]. Following lemma is frequently used in the main results.

Lemma 1.1

[9] For 1 ≤ i ≤ n and 1 ≤ j ≤ 2, assume that <d11,d21,d31,...,dn1> and <d12,d22,d32,...,dn2> are degree sequences with the condition of di1=di2, where dij is degree of the vertex vij∈V(Gj) and n = |V(G1)| = |V(G2)|. Then, F(G1) = F(G2).

2 Computational results of F-index

A connected graph with order n and size m such that m = n – 1 + c is called a c-cyclic graph. In particular, if c = 0, c = 1, c = 2 or c = 3 then it is a tree, unicyclic, bicyclic or tricyclic graph respectively. A tricyclic graph contains at least three and at most seven cycles except of exactly five cycles. There are seven possibilities for a tricyclic graph with three cycles as shown in Figure 1. Moreover, the possibilities for the tricyclic graphs with four, six and seven cycles are four, three and one respectively, see Figure 2. Now, we define some more tricyclic graphs with respect to the attachment of k ≥ 1 pendent vertices to the l vertices of the graphs which are defined in Figure 1. To choose l vertices, we have the following choices:

Figure 1 Tricyclic graphs with three cycles.
Figure 1

Tricyclic graphs with three cycles.

Figure 2 Tricyclic graphs with four cycles (H1, H2 H3 and H4), six cycles (L1, L2 and L3) and seven cycles (K).
Figure 2

Tricyclic graphs with four cycles (H1, H2 H3 and H4), six cycles (L1, L2 and L3) and seven cycles (K).

  1. cycle-vertex of degree 2,

  2. tree-vertex of degree 2,

  3. cycle-vertex of degree greater or equal to 2,

  4. cycle-vertex and tree-vertex of degree exactly 2,

  5. cycle-vertex of degree greater or equal to 2 and tree-vertex of degree exactly 2.

More precisely, we define that l vertices are either of degree exactly 2 or, greater or equal to 2. By joining k ≥ 1 pendant vertices to l vertices of degree 2, and the vertices of degree greater or equal to 2 of the graph G1 in Figure 1, the tricyclic graphs Al,k,1m,r=A1 and Al,k,2m,r=A2 are obtained respectively. In G1, vertices of degree 3 are four and of degree 2 are m1 + m2 + m3 + r such that m = m1 + m2 + m3 are cycle-vertex and r are tree-vertex. Table 1 shows the vertex-partition with respect to degrees of vertices of graphs A1 and A2.

Table 1

Vertex-partitions of the tricyclic graphs A1 and A2.

d(v), for v ∈ V(A1) 1 2 3 k + 2
|d(v)| lk m + r – l 4 l
d(v), for v ∈ V(A2) 1 2 k + 2 k + 3
|d(v)| lk m + r + 4 – l l – 4 4

In G2 (Figure 1), the vertices of degrees 4, 3, and 2 are 1, 2 and m1 + m2 + m3 + r + 1 such that m = m1 + m2 + m3 are cycle-vertex and r + 1 are tree-vertex. By joining k ≥ 1 pendant vertices to l vertices of degree 2, and the vertices of degree greater or equal to 2 of G2 in Figure 1, the tricyclic graphs Bl,k,1m,r=B1 and Bl,k,2m,r=B2, are obtained, respectively. The Table 2 presents the vertex-partitions of graphs B1 and B2.

Table 2

Vertex-partitions of the tricyclic graphs B1 and B2.

d(v), for v ∈ V(B1) 1 2 3 4 k + 2
|d(v)| lk m + r + 1 – l 2 1 l
d(v), for v ∈ V(B2) 1 2 k + 2 k + 3 k + 4
|d(v)| lk m + r + 4 – l l – 3 2 1

The graph G3 (Figure 1) has 2 and m1 + m2 + m3 + r + 2 vertices of degrees 4 and 2, respectively such that m = m1 + m2 + m3 and r + 2 are cycle-vertex. The tricyclic graphs Cl,k,1m,r=C1 and Cl,k,2m,r=C2 are obtained by joining k ≥ 1 pendent vertices to l vertices of degree 2, and degree greater or equal to 2 of the graph G3 in Figure 1, respectively. The Table 3 presents the vertex-partitions with respect to the degrees of vertices of the graphs C1 and C2.

Table 3

Vertex-partitions of the tricyclic graphs C1 and C2.

d(v), for v ∈ V(C1) 1 2 4 k + 2
|d(v)| lk m + r + 2 – l 2 l
d(v), for v ∈ V(C2) 1 2 k + 2 k + 4
|d(v)| lk m + r + 4 – l l – 2 2

Similarly, we obtain the tricyclic graphs Dl,k,1m,r=D1,Dl,k,2m,r=D2,El,k,1m,r=E1 and El,k,2m,r=E2, by joining k ≥ 1 pendent vertices to l vertices of degree 2, and greater or equal to 2 of the graphs G4 and G5 in Figure 1, respectively. In Figure 1, G4 has m cycle-vertex and r + 2 tree-vertex of degrees 2 and G5 has m cycle-vertex and r + 3 tree-vertex of degrees 2. The vertex-partitions of these derived tricyclic graphs are shown in Table 4 and Table 5.

Table 4

Vertex-partitions of the tricyclic graphs D1 and D2.

d(v), for v ∈ V(D1) 1 2 3 5 k + 2
|d(v)| lk m + r + 2 – l 1 1 l
d(v), for v ∈ V(D2) 1 2 k + 2 k + 3 k + 5
|d(v)| lk m + r + 4 – l l – 2 1 1

Table 5

Vertex-partitions of the tricyclic graphs E1 and E2.

d(v), for v ∈ V(E1) 1 2 6 k + 2
|d(v)| lk m + r + 3 – l 1 l
d(v), for v ∈ V(E2) 1 2 k + 2 k + 6
|d(v)| lk m + r + 4 – l l – 1 1

Moreover for i ∈ {1, 2}, we note that (i) |V(Ai)| = |V(Bi)| = |V(Ci)| = |V(Di)| = |V(Ei)| = m1 + m2 + m3 + lk + r + 4 = m + lk + r + 4 and (ii) the graphs G6 and G7 have the same degree sequences as of G1 and G2 respectively. For more explanation, B1, B2, E1 and E2 are given in Figure 2 with certain value of the parameters l, m, k and r.

Now, A11 from A1 are obtained by deleting k pendant vertices from a vertex of degree k + 2 and joining these vertices to another vertex of degree k + 2. Similarly, A12 is derived from A11 by deleting 2k pendant vertices from the vertex of degree 2k + 2 and joining these vertices to the vertex of degree k + 2. After l – 1 iteration, we obtain A1l−1 from A1l−2 by deleting (l – 1)k pendent vertices from a vertex of degree (l – 1)k + 2 and joining these vertices to the last vertex of degree k + 2, where 2 ≤ l ≤ m + r. Using the same transformation, we obtain A2i from A2i−1 for 1 ≤ i ≤ l – 5 by the deletion of ik pendent vertices from a vertex of degree ik + 2 and joining these vertices to the vertex of degree k + 2. Moreover, we obtain A2i from A2i−1 for l – 4 ≤ i ≤ l – 1 by the deletion of ik pendent vertices from a vertex of degree ik + 3 and joining these vertices to the last vertex of degree k + 3, where A20 = A2. Similarly, for 1 ≤ i ≤ l – 1, we obtain B2i,C2i,D2i and E2i from B2i−1,C2i−1,D2i−1, and E2i−1 respectively.

Assume that 𝓤1, 𝓤2, 𝓤3 𝓤4 and 𝓤5, are classes of the tricyclic graphs obtained from G1, G2, G3 G4 and G5 (shown in Figure 1) respectively such that the order of each graph is m + lk + r + 4 with lk pendent vertices. Let Unlk be a class of all the tricyclic graphs with three cycles such that each graph has order n and pendant vertices lk, where k ≥ 1, n ≥ 16 and 1 ≤ l ≤ n. Similarly, tricyclic graphs with four and six cycles obtained from the base graphs presented in Figure 2 are given in Table 6 and Table 7. Moreover, ξnlk and ζnlk are classes of all the tricyclic graphs with four and six cycles respectively that include each graph of order n and pendant vertices lk. Finally, we obtain the tricyclic graphs with seven cycles Kl,k,1m,r=K1) and (Kl,k,2m,r=K2) from the base graph K (see, Figure 2) and μnlk be a class of all the tricyclic graphs with seven cycles such that each graph has order n and pendant vertices lk. Now, by the deletion and addition of pendant vertices, we have Rji,Sji,Tji,Xji,Yji,Zji and Kji from Rji−1,Sji−1,Tji−1,Xji−1,Yji−1,Zji−1 and Kji−1 respectively, where 1 ≤ i ≤ l – 1 and 1 ≤ j ≤ 2.

Table 6
Base Graphs (BG) H1 H2 H3
Joining k vertices to l vertices of degree = 2 Rl,k,1m,r = R1 Sl,k,1m,r = S1 Tl,k,1m,r = T1
Joining k vertices to l vertices of degree ≥ 2 Rl,k,2m,r = R2 Sl,k,2m,r = S2 Tl,k,2m,r = T2
Classes of tricyclic graphs generated from BG ξ1 ξ2 ξ3

Table 7
Base Graphs (BG) L1 L2 L3
Joining k vertices to l vertices of degree = 2 Xl,k,1m,r = X1 Yl,k,1m,r = Y1 Zl,k,1m,r = Z1
Joining k vertices to l vertices of degree ≥ 2 Xl,k,2m,r = X2 Yl,k,2m,r = Y2 Zl,k,2m,r = Z2
Classes of tricyclic graphs generated from BG ζ1 ζ2 ζ3

Now, we present some important lemmas which are frequently used in the next section of main results.

Lemma 2.1

For u, v, a, b ≥ 1, the functions (i) f1(u) = –au(u + b), (ii) f2(u) = –au2(u + b), (iii) f3(u, v) = –3u3(v2 + 3v + 2) – 12u2(v + 1), and (iv) f4(u, v) = –auv(uv + b) – c are strictly decreasing functions.

Proof

Since, (i) df1(u)du = –a(2u + b) < 0, (ii) df2(u)du = –au(3u + 2b) < 0, (iii) ∂f3(u,v)∂u = –9u2(v2 + 3v + 2) – 24u(v + 1) < 0 and ∂f3(u,v)∂v = –3u3(2v + 3) – 12u2 < 0, (iv) ∂f4(u,v)∂u = –av(2uv + b) < 0 and ∂f4(u,v)∂v = –au(2uv + b) < 0 for u, v, a, b ≥ 1. Therefore, f1(u), f2(u), f3(u, v) and f4(u, v) are strictly decreasing functions.

Figure 3 (i)B4,2,16,2(ii)B3,2,23,2(iii)E4,2,17,0 and (iv)E3,2,24,0. $\begin{array}{} \displaystyle (\text{i})B^{6,2}_{4,2,1} (\text{ii})B^{3,2}_{3,2,2} (\text{iii})E^{7,0}_{4,2,1} ~\mbox{and}~ (\text{iv})E^{4,0}_{3,2,2}. \end{array}$
Figure 3

(i)B4,2,16,2(ii)B3,2,23,2(iii)E4,2,17,0 and (iv)E3,2,24,0.

By the use of Definition 1.3 and the generalization of Table 1-Table 5 for the ith iteration of the deletion of ik pendant vertices from the vertex of degree ik + 2 and joining them to a vertex of degree k + 2, we obtain the F-index of the tricyclic graphs A1i,B1i,C1i,D1i and E1i with three cycles and lk pendant vertices for 0 ≤ i ≤ l – 1 in the following lemma.

Lemma 2.2

For m = m1 + m2 + m3 ≥ 9, k ≥ 1, r ≥ 2, 2 ≤ l ≤ m + r, 0 ≤ i ≤ l – 1 and mj ≥ 3 with j = 1, 2, 3, the F-index of the tricyclic graphs defined above are

(i)aF(A1i)=lk+(l−i−1)(k+2)3+8(m+r−l+i)+[(i+1)k+2]3+108,
(ii)aF(B1i)=lk+(l−i−1)(k+2)3+8(m+r+1−l+i)+[(i+1)k+2]3+118,
(iii)aF(C1i)=lk+(l−i−1)(k+2)3+8(m+r−l+2+i)+[(i+1)k+2)]3+128,
(iv)aF(D1i)=lk+(l−i−1)(k+2)3+8(m+r−l+2+i)+[(i+1)k+2)]3+152,
(v)aF(E1i)=lk+(l−i−1)(k+2)3+8(m+r−l+3+i)+[(i+1)k+2)]3+216.

Again using Definition 1.3 and Tables 1-5 (3rd and 4th rows), we obtain the F-index of A2i,B2i,C2i,D2i and E2i for 0 ≤ i ≤ l – 1 in the following lemma.

Lemma 2.3

For m = m1 + m2 + m3 ≥ 9, k ≥ 1, r ≥ 2, 2 ≤ l ≤ m + r + 4, 0 ≤ i ≤ l – 1 and mj ≥ 3 with j = 1, 2, 3, the F-index of the tricyclic graphs A2i,B2i,C2i,D2i and E2i are

(i)aF(A2i)=lk+(l−5−i)(k+2)3+4(k+3)3+8(m+r+4−l+i)+[(i+1)k+3]3;fora1≤i≤l−5,lk+(l−1−i)(k+3)3+8(m+r)+27(i−l+4)+[(i+1)k+3]3;foral−4≤i≤l−1,
(ii)aF(B2i)=lk+(l−4−i)(k+2)3+2(k+3)3+(k+4)3+8(m+r+4−l+i)+[(i+1)k+3]3;fora1≤i≤l−4,lk+8(m+r+1)+(k+3)3+(k+4)3+(lk−2k+3)3;forai=l−3,lk+8(m+r+1)+(k+4)3+(lk−k+3)3+27;forai=l−2,lk+(lk+4)3+8(m+r+1)+54;forai=l−1,
(iii)aF(C2i)=lk+(l−3−i)(k+2)3+2(k+4)3+8(m+r+4−l+i)+[(i+1)k+3]3;  fora1≤i≤l−3,lk+(k+4)3+8(m+r+2)+(lk−k+4)3;aforai=l−2,lk+(lk+4)3+8(m+r+2)+64;aforai=l−1,
(iv)aF(D2i)=lk+(l−3−i)(k+2)3+(k+3)3+(k+5)3+8(m+r+4−l+i)+[(i+1)k+3]3; fora1≤i≤l−3,lk+(k+5)3+(lk−k+3)3+8(m+r+2);aforai=l−2,lk+(lk+5)3+8(m+r+2)+27;aforai=l−1,
(v)aF(E2i)=lk+(l−2−i)(k+2)3+(k+6)3+8(m+r+4−l+i)+[(i+1)k+3]3;fora1≤i≤l−2,lk+(lk+6)3+8(m+r+3);forai=l−1.

Lemma 2.4

For m = m1 + m2 + m3 ≥ 9, k ≥ 1, r ≥ 2, 2 ≤ l ≤ m + r, 0 ≤ i ≤ l – 1, 1 ≤ j ≤ 2 and mp ≥ 3 with p = 1, 2, 3, we have

  1. F( Tji ) = F( Aji ), F( Sji ) = F( Bji ) and F( Rji ) = F( Dji )

  2. F( Xji ) = F( Cji ), F( Yji ) = F( Bji ) and F( Zji ) = F( Aji )

  3. F( Lji ) = F( Aji ).

Proof

(a) Since the degree sequences of the base graphs of the tricyclic graph with four cycles Tji (see, H3 in Figure 2) and tricyclic graph with three cycles Aji (see, G1 in Figure 1) are equal. Therefore, the degree sequences of the graphs Tji and Aji having k ≥ 1 pendant vertices attached with l vertices of degree (i) exactly 2 for j = 1 and (ii) greater or equal 2 for j = 2 are equal. Consequently, by Lemma 1.1, F( Tji ) = F( Aji ). Similarly, the degree sequences of the tricyclic graphs with four cycles Sji and Rji are equal to the degree sequences of the tricyclic graphs with three cycles Bji and Dji respectively. Thus, by Lemma 1.1, we have F( Sji ) = F( Bji ) and F( Rji ) = F( Dji ). (b) Proof is same as of part (a). (c) Proof is same as of part (a).

3 Extremal graphs with respect to F-index

The results of extremal graphs in the complete class of tricyclic graphs with fixed pendant vertices are obtained in this section.

Lemma 3.1

For m = m1 + m2 + m3 ≥ 9, k ≥ 1, r ≥ 2, 2 ≤ l ≤ m + r and mj ≥ 3 with j = 1, 2, 3. Then, F(A1) ≤ F(A2), F(B1) ≤ F(B2), F(C1) ≤ F(C2), F(D1) ≤ F(D2) and F(E1) ≤ F(E2).

Proof

Consider:

  1. Using Lemma 2.2(i) and Lemma 2.3(i) for i = 0, we have F(A1)–F(A2) = F(Al,k,lm,r)−F(Al,k,2m,r) = –12k(k + 5). By Lemma 2.1(i), it follows that the tricyclic graph A1 has F-index less than of A2.

  2. Using Lemma 2.2(ii) and Lemma 2.3(ii) for i = 0, we have F(B1) – F(B2) = F(Bl,k,lm,r)−F(Bl,k,2m,r) = –6k(2k + 11). By Lemma 2.1(i), it follows that F-index of the tricyclic graph B1 is less than the F-index of the tricyclic graph B2.

  3. Using Lemma 2.2(iii) and Lemma 2.3(iii) for i = 0, we have F(C1) – F(C2) = F(Cl,k,lm,r)−F(Cl,k,2m,r) = –12k(k + 6). By Lemma 2.1(i), it follows that F-index of the tricyclic graph C1 is less than the F-index of the tricyclic graph C2.

  4. Using Lemma 2.2(iv) and Lemma 2.3(iv) for i = 0, we have F(D1) – F(D2) = F(Dl,k,lm,r)−F(Dl,k,2m,r) = –12k(k + 6). By Lemma 2.1(i), it follows that F-index of the tricyclic graph D1 is less than the F-index of the tricyclic graph D2.

  5. Using Lemma 2.2(v) and Lemma 2.3(v) for i = 0, we have F(E1) – F(E2) = F(El,k,lm,r)−F(El,k,2m,r) = –12k(k + 8). By Lemma 2.1(i), it follows that F-index of the tricyclic graph E1 is less than the F-index of E2.

Consequently, from all the cases, we have F(A1) ≤ F(A2), F(B1) ≤ F(B2), F(C1) ≤ F(C2), F(D1) ≤ F(D2) and F(E1) ≤ F(E2).

Lemma 3.2

For m = m1 + m2 + m3 ≥ 9, k ≥ 1, r ≥ 2, 2 ≤ l ≤ m + r and mj ≥ 3 with j = 1, 2, 3. Then, F(A1) ≤ F(B1) ≤ F(C1) ≤ F(D1) ≤ F(E1).

Proof

Consider:

  1. By Lemma 2.2((i) and (ii)) for i = 0, we have F(A1) – F(B1) = F(Al,k,lm,r)−F(Bl,k,1m,r) = –18 < 0. It follows that F(A1) < F(B1).

  2. Using Lemma 2.2((ii) and (iii)) for i = 0, we have F(B1) – F(C1) = F(Bl,k,lm,r)−F(Cl,k,1m,r) = –36 < 0. Which implies that F(B1) < F(C1).

  3. By Lemma 2.2((iii) and (iv)) for i = 0, we have F(C1) – F(D1) = F(Cl,k,lm,r)−F(Dl,k,1m,r) = –24 < 0. It follows that F(C1) < F(D1).

  4. By Lemma 2.2((iv) and (v)) for i = 0, we have F(D1) – F(E1) = F(Dl,k,lm,r)−F(El,k,1m,r) = –72 < 0. It implies that F(D1) < F(E1).

From all the cases, we conclude that F(A1) ≤ F(B1) ≤ F(C1) ≤ F(D1) ≤ F(E1).

Theorem 3.3

If m = m1 + m2 + m3 ≥ 9, k ≥ 1, r ≥ 2, 2 ≤ l ≤ m + r and mj ≥ 3 with j = 1, 2, 3. Then, F(A1) ≤ F(G), F(B1) ≤ F(G), F(C1) ≤ F(G), F(D1) ≤ F(G) and F(E1) ≤ F(G) for each G ∈ 𝓤1, G ∈ 𝓤2, G ∈ 𝓤3, G ∈ 𝓤4 and G ∈ 𝓤5 respectively. Moreover, equalities hold if G ≅ A1, G ≅ B1, G ≅ C1, G ≅ D1 and G ≅ E1 respectively.

Proof

We consider the following cases:

  1. If G = A11 then by Lemma 2.2(i) (for i = 0 and i = 1) F(A1) – F( A11 ) = –6k2(k + 2). Moreover using Lemma 2.1(ii), we have F(A1) < F( A11 ). Similarly, by Lemma 2.2((ii)-(iv)) and Lemma 2.1(ii), we have F(B1) – F( B11 ) = F(C1) – F( C11 ) = F(D1) – F( D11 ) = F(E1) – F( E11 ) = –6k2(k + 2) < 0 which implies that F(B1) ≤ F( B11 ), F(C1) ≤ F( C11 ), F(D1) ≤ F( D11 ), F(E1) ≤ F( E11 ). Consequently, if G = A11 , G = B11 , G = C11 , G = D11 and G = E11 then F(A1) ≤ F(G), F(B1) ≤ F(G) F(C1) ≤ F(G), F(D1) ≤ F(G) and F(E1) ≤ F(G) for each G ∈ 𝓤1, G ∈ 𝓤2, G ∈ 𝓤3, G ∈ 𝓤4 and G ∈ 𝓤5 respectively.

  2. If G = A1i for 1 ≤ i ≤ l – 2, by Lemma 2.2(i)

    F(A1i)−F(A1i+1)=−8+(k+2)3+(ik+k+2)3−(ik+2k+2)3=−3k3(i2+3i+2)−12k2(i+1).

    By Lemma 2.1(iii), F(A1i)<F(A1i+1). Using i = 1, 2, 3, ..., l – 2, we have F(A11)  <  F(A12),F(A12)  <  F(A13),...,F(A1l−2)<F(A1l−1). By Combining these inequalities

    F(A11)<F(A12)<F(A13)<...<F(A1l−1).

    Using Case 1 and above inequality, F(A1) < F( A1i ) for 1 ≤ i ≤ l – 1 which implies that F(A1) < F(G). Similarly, by Lemma 2.2((ii)-(v)) F(B1i)−F(B1i+1)=F(C1i)−F(C1i+1)=F(D1i)−F(D1i+1)=F(E1i)−F(E1i+1) = –8 + (k + 2)3 + [k(i + 1) + 2]3 – [k(i + 2) + 2)]3 = –3k3(i2 + 3i + 2) – 12k2(i + 1). By Lemma 2.1(ii), F(B1i)<F(B1i+1),F(C1i)<F(C1i+1),F(D1i)<F(D1i+1) and F(E1i)<F(E1i+1), where 1 ≤ i ≤ l – 2. Using Case 1 and above inequalities, we have F(B1)<F(B1i),F(C1)<F(C1i),F(D1)<F(D1i) and F(E1)<F(E1i), , for 1 ≤ i ≤ l – 1. Consequently, if G=A1i,G=B1i,G=C1i,G=D1i and G=E1i for 1 ≤ i ≤ l – 1 then F(A1) ≤ F(G), F(B1) ≤ F(G) F(C1) ≤ F(G), F(D1) ≤ F(G) and F(E1) ≤ F(G) for each G ∈ 𝓤1, G ∈ 𝓤2, G ∈ 𝓤3, G ∈ 𝓤4 and G ∈ 𝓤5 respectively.

  3. If G=A2i,G=B2i,G=C2i,G=D2i and G=E2i, using the same way as of Case 2, we can prove that F(A2) ≤ F(G), F(B2) ≤ F(G) F(C2) ≤ F(G), F(D2) ≤ F(G) and F(E2) ≤ F(G) respectively, where 1 ≤ i ≤ l – 1. Moreover, by Lemma 3.1 F(A1) ≤ F(A2), F(B1) ≤ F(B2), F(C1) ≤ F(C2), F(D1) ≤ F(D2) and F(E1) ≤ F(E2). Consequently, F(A1) ≤ F(G), F(B1) ≤ F(G) F(C1) ≤ F(G), F(D1) ≤ F(G) and F(E1) ≤ F(G) for each G ∈ 𝓤1, G ∈ 𝓤2, G ∈ 𝓤3, G ∈ 𝓤4 and G ∈ 𝓤5 respectively.

  4. If G∈U1−{A1i,A2i},G∈U2−{B1i,B2i},G∈U3−{C1i,C2i},G∈U4−{D1i,D2i} and G∈U5−{E1i,E2i}. After using transformation of the deletion of pendant vertices and joining them with degree greater or equal to two, we obtain A1i or A2i,B1i or B2i,C1i or C2i,D1i or D2i , and E1i or E2i respectively. Thus, we follow the Case 2 or Case 3.

    Thus, from all the cases, we have F(A1) ≤ F(G) for each G ∈ 𝓤1, F(B1) ≤ F(G) for each G ∈ 𝓤2, F(C1) ≤ F(G) for each G ∈ 𝓤3, F(D1) ≤ F(G) for each G ∈ 𝓤4 and F(E1) ≤ F(G) for each G ∈ 𝓤5, where equalities hold if G ≅ A1, G ≅ B1, G ≅ C1, G ≅ D1 and G ≅ E1 respectively.

Theorem 3.4

If k ≥ 1, n ≥ 16 + lk and 1 ≤ l ≤ n – lk. Then F(A1) ≤ F(G) for each G ∈ Unlk , where Unlk is a class of all the tricyclic graphs with three cycles such that each graph has order n and pendant vertices lk. Moreover, equality holds if G ≅ A1.

Proof

We consider the following cases:

  1. If G ∈ 𝓤1, by Theorem 3.3 F(A1) ≤ F(G) for each G ∈ 𝓤1.

  2. If G ∈ 𝓤2, by Theorem 3.3 F(B1) ≤ F(G) for each G ∈ 𝓤2. Also, by Lemma 3.6 F(A1) ≤ F(B1) which implies that F(A1) ≤ F(G) for each G ∈ 𝓤2.

  3. Assume that G ∈ 𝓤3. Using Theorem 3.3, we have F(C1) ≤ F(G) for each G ∈ 𝓤3. Now, by Lemma 3.2 F(A1) ≤ F(C1). Consequently, F(A1) ≤ F(G) for each G ∈ 𝓤3.

  4. If G ∈ 𝓤4 then by Theorem 3.3 F(D1) ≤ F(G) for each G ∈ 𝓤4. Moreover, by Lemma 3.2 F(A1) ≤ F(D1) which implies that F(A1) ≤ F(G) for each G ∈ 𝓤4.

  5. If G ∈ 𝓤5, by Theorem 3.3 F(C1) ≤ F(G) for each G ∈ 𝓤5 and by Lemma 3.2 F(A1) ≤ F(E1). Consequently, F(A1) ≤ F(G) for each G ∈ 𝓤5.

    So, from all the cases, we conclude that F(A1) ≤ F(G) for each G ∈ Unlk and equality holds if G ≅ A1.

Lemma 3.5

For m = m1 + m2 + m3 ≥ 9, k ≥ 1, r ≥ 2, 2 ≤ l ≤ n – lk and mj ≥ 3 with j = 1, 2, 3. Then, F( F(A1l−1)≤F(A2l−1),F(B1l−1)≤F(B2l−1),F(C1l−1)≤F(C2l−1),F(D1l−1)≤F(D2l−1) and F(E1l−1)≤F(E2l−1).

Proof

Consider:

  1. By Lemma 2.2(i) and Lemma 2.3(i) for i = l – 1, we have F( A1l−1 ) – F( A2l−1 ) = –3lk(lk + 5). By Lemma 2.1(iv), F( A1l−1 ) < F( A2l−1 ).

  2. Using Lemma 2.2(ii) and Lemma 2.3(ii) for i = l – 1, we have F( B1l−1 ) – F( B2l−1 ) = –6lk(lk + 6). Using Lemma 2.1(iv), we have F( B1l−1 ) ≤ F( B2l−1 ).

  3. By Lemma 2.2(iii) and Lemma 2.3(iii) for i = l – 1, we have F( C1l−1 ) – F( C2l−1 ) = –6lk(lk + 6). By Lemma 2.1(iv), F( C1l−1 ) < F( C2l−1 ).

  4. By Lemma 2.2(iv) and Lemma 2.3(iv) for i = l – 1, we have F( D1l−1 ) – F( D2l−1 ) = –9lk(lk + 7). By Lemma 2.1(iv), F( D1l−1 ) < F( D2l−1 ).

  5. By Lemma 2.2(v) and Lemma 2.3(v) for i = l – 1, we have F( E1l−1 ) – F( E2l−1 ) = –12lk(lk + 8). Using Lemma 2.1(iv), F( E1l−1 ) < F( E2l−1 ).

    From all the cases, we conclude that F( A1l−1 ) ≤ F( A2l−1 ), F( B1l−1 ) ≤ F( B2l−1 ), F( C1l−1 ) ≤ F( C2l−1 ), F( D1l−1 ) ≤ F( D2l−1 ), F( E1l−1 ) ≤ F( E2l−1 ).

Lemma 3.6

For m = m1 + m2 + m3 ≥ 9, k ≥ 1, r ≥ 2, 2 ≤ l ≤ m + r + 4 and mj ≥ 3 with j = 1, 2, 3. Then, F( A2l−1 ) ≤ F( B2l−1 ) ≤ F( C2l−1 ) ≤ F( D2l−1 ) ≤ F( E2l−1 ).

Proof

Consider:

  1. By Lemma 2.3((i) and (ii)) for i = l – 1, we have F( A2l−1 ) – F( B2l−1 ) = –3l2k2 – 21lk – 28. Using Lemma 3.1(iv), we have F( A2l−1 ) < F( B2l−1 ).

  2. Using Lemma 2.3((ii) and (iii)) for i = l – 1, we have F( B2l−1 ) – F( C2l−1 ) = –18 < 0. By Lemma 3.1(iv), we have F( B2l−1 ) ≤ F( C2l−1 ).

  3. By Lemma 2.3((iiii) and (iv)) for i = l – 1, we have F( C2l−1 ) – F( D2l−1 ) = –3l2k2 – 27lk – 24. Using Lemma 3.1(iv), we have F( C2l−1 ) < F( D2l−1 ).

  4. By Lemma 2.3((iv) and (v)) for i = l – 1, we have F( D2l−1 ) – F( E2l−1 ) = –3l2k2 – 33lk – 72. By Lemma 3.1, we have F( D2l−1 ) < F( E2l−1 ).

    From all the cases, we conclude that F( A2l−1 ) ≤ F( B2l−1 ) ≤ F( C2l−1 ) ≤ F( D2l−1 ) ≤ F( E2l−1 ).

Theorem 3.7

If m = m1 + m2 + m3 ≥ 9, k ≥ 1, r ≥ 2, 2 ≤ l ≤ m + r + 4 and mj ≥ 3 with j = 1, 2, 3. Then, F(G) ≤ F( A2l−1 ), F(G) ≤ F( B2l−1 ), F(G) ≤ F( C2l−1 ), F(G) ≤ F( D2l−1 ) and F(G) ≤ F( E2l−1 ) for each G ∈ 𝓤1, G ∈ 𝓤2, G ∈ 𝓤3, G ∈ 𝓤4 and G ∈ 𝓤5 respectively. Moreover, equalities hold if G ≅ A2l−1 , G ≅ B2l−1 , G ≅ C2l−1 , G ≅ D2l−1 and G ≅ E2l−1 respectively.

Proof

Proof is same as of Theorem 3.3 with the help of Lemma 2.3, Lemma 3.5 and Lemma 3.6.

Theorem 3.8

If k ≥ 1, n ≥ 16 + lk and 1 ≤ l ≤ n – lk. Then F(G) ≤ F( E2l−1 ) for each G ∈ Unlk , where Unlk is a class of all the tricyclic graphs with three cycles such that each graph has order n and pendant vertices lk. Moreover, equality holds if G ≅ E2l−1 .

Proof

Proof follows by Theorem 3.4 with the help of Theorem 3.3, Lemma 3.5 & Lemma 3.6.

By the similar arguments as of the tricyclic graphs with three cycles, we obtain the following result for the tricyclic graphs with four, six and seven cycles.

Theorem 3.9

If k ≥ 1, n ≥ 16 + lk and 1 ≤ l ≤ n – lk. Then

  1. F(T1) ≤ F(G) ≤ F( R2l−1 ) for each G ∈ ξnlk , where lower and upper bounds holds for G ≅ T1 and G ≅ R2l−1 respectively.

  2. F(Z1) ≤ F(G) ≤ F( X2l−1 ) for each G ∈ ζnlk , where lower and upper bounds holds for G ≅ Z1 and G ≅ X2l−1 respectively.

  3. F(L1) ≤ F(G) ≤ F( L2l−1 ) for each G ∈ μnlk , where lower and upper bounds holds for G ≅ L1 and G ≅ L2l−1 respectively.

Proof

(a) By Lemma 2.4, F(T1) = F(A1), F(T2) = F(A2), F(S1) = F(B1), F(S2) = F(B2), F(R1) = F(D1) and F(R2) = F(D2). Now, by Lemma 3.1 F(A1) ≤ F(A2), F(B1) ≤ F(B2) and F(D1) ≤ F(D2). Consequently, F(T1) ≤ F(T2), F(S1) ≤ F(S2) and F(R1) ≤ F(R2). Moreover, by Lemma 3.2 F(A1) ≤ F(B1) ≤ F(D1) which implies that F(T1) ≤ F(S1) ≤ F(R1). Finally, by Theorem 3.3 and Theorem 3.4, we have F(A1) ≤ F(G) for each G ∈ Unlk , where Unlk is a class of all the tricyclic graphs with three cycles such that each graph has order n and pendant vertices lk. Consequently, F(T1) ≤ F(G) for each G ∈ ξnlk , where lower bound holds for G ≅ T1. Similarly, by Theorem 3.5, Lemma 3.6, Theorem 3.7 and Theorem 3.8, we have F(G) ≤ F( D2l−1 ) ≤ F( E2l−1 ) for each G ∈ Unlk , where Unlk is a class of all the tricyclic graphs with three cycles such that each graph has order n and pendant vertices lk which implies that F(G) ≤ F( R2l−1 ) for each G ∈ ξnlk , where ξnlk is a class of all the tricyclic graphs with four cycles such that each graph has order n and pendant vertices lk. Thus, F(T1) ≤ F(G) ≤ F( R2l−1 ) for each G ∈ ξnlk , where lower and upper bounds holds for G ≅ T1 and G ≅ R2l−1 respectively. (b) Proof is same as of part (a). (c) Proof is same as of part (a).

Theorem 3.10

If k ≥ 1, n ≥ 16 + α and 1 ≤ l ≤ n – α. Then F(A1) ≤ F(G) ≤ F( E2l−1 ) for each G ∈ Ωnα , where Ωnα={Unα,ξnα,ζnα,μnα}, α is number of pendant vertices and bounds (lower and upper) holds for G ≅ T1 and G ≅ R2l−1 (respectively).

4 Lower and upper bounds

The ordering and investigate of bounds (lower and upper) of the F-index in the complete class of tricyclic graphs of three, four, six or seven cycles with fixed pendant vertices is given in this section.

Theorem 4.1

If m = m1 + m2 + m3 ≥ 9, k ≥ 1, r ≥ 2, 2 ≤ l ≤ m + r + 4 and mj ≥ 3 with j = 1, 2, 3. Then, (a) F(𝓤1) < F(𝓤2) < F(𝓤3) < F(𝓤4) < F(𝓤5), (b)F(ξ3) < F(ξ2) < F(ξ1) and (c)F(ζ3) < F(ζ2) < F(ζ1).

Proof

(a) Firstly, we prove that F(𝓤1) < F(𝓤2). For the purpose, we show that for each G ∈ 𝓤1 there exists G∗ ∈ 𝓤2 such that F(G) < F(G∗), where n = m + r + 4 + lk is order of both G and G∗ with lk pendant vertices in each. We assume that G = A1i and G∗ = B1i for 1 ≤ i ≤ l – 1. By Lemma 2.2, F(G) – F(G∗) = –10 < 0 which implies that F(G) < F(G∗). Similarly, by Lemma 2.3 it can be proved that if G = A2i for 1 ≤ i ≤ l – 1 then there exists G∗ = B2i such that F(G) < F(G∗). In addition, if G∈U1−{A1i,A2i}, using transformation of delation and joining the pendant vertices. Then, G = A1i or G = A2i . Thus, we get G∗ ∈ 𝓤2 such that F(G) < F(G∗). So, we conclude that F(𝓤1) < F(𝓤2). Similarly, we can prove that F(𝓤2) < F(𝓤3), F(𝓤3) < F(𝓤4), and F(𝓤4) < F(𝓤5). Consequently, we have F(𝓤1) < F(𝓤2) < F(𝓤3) < F(𝓤4) < F(𝓤5). Proves of (b) and (c) are same as of (a).

Theorem 4.2

If m = m1 + m2 + m3 ≥ 9, k ≥ 1, r ≥ 2, 2 ≤ l ≤ m + r + 4 and mj ≥ 3 with j = 1, 2, 3. Then, F(𝓤1) = F(ξ3) = F(ζ3) < F(𝓤2) = F(ξ2) = F(ζ2) < F(𝓤3) = F(ζ1) < F(𝓤4) = F(ξ1) < F(𝓤5) < F(𝓤6).

Proof

Proof is obvious using Lemma 2.4, Theorem 4.1 (a), (b) and (c).

Theorem 4.3

Let G ∈ Ωnα be a tricyclic graph of order n ≥ 16 + α with three, four, six or seven cycles and α ≥ 1 pendant vertices. Then, 8n + 12α + 76 ≤ F(G) ≤ 8(n – 1) – 7α + (α + 6)3, where the lower and upper bounds are achieved if and only if G ≅ A1 with k = 1 and G ≅ E2l−1 respectively.

Proof

Assuming l = α and k = 1 in Lemma 2.2 (i) for i = 0 and Lemma 2.3(v) for i = l – 1, we have F(A1) = 8n + 12α + 76 and F( E2l−1 ) = 8(n – 1) – 7α + (α + 6)3. Moreover, by Theorem 3.4 and Theorem 3.8 F(A1) < F(G) and F(G) ≤ F( E2l−1 ) for each G being a tricyclic graph of order n ≥ 16 with three cycles and α ≥ 1 pendant vertices. Thus, we have 8n + 12α + 76 ≤ F(G) ≤ 8(n – 1) – 7α + (α + 6)3, where lower and upper bounds are achieved if and only if G ≅ A1 with k = 1 and G ≅ E2l−1 respectively.

5 Conclusion

In this paper, we studied the complete class of tricyclic graphs consisting on three, four, six and seven cycles for certain number of pendant vertices with respect to F-index. We proved the existence of the extremal graphs and construct the ordering of graphs with respect to F-index. Mainly, we computed the bounds (lower and upper) of F-index for the same family of graphs.

Acknowledgement

The authors are indebted to the anonymous referees for their valuable comments to improve the original version of this paper.

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Received: 2018-09-10
Accepted: 2020-01-18
Published Online: 2020-03-13

© 2020 Sana Akram et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  60. Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability
  61. Asymptotic normality and mean consistency of LS estimators in the errors-in-variables model with dependent errors
  62. Some non-commuting solutions of the Yang-Baxter-like matrix equation
  63. General (p,q)-mixed projection bodies
  64. An extension of the method of brackets. Part 2
  65. A new approach in the context of ordered incomplete partial b-metric spaces
  66. Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis
  67. Remark on subgroup intersection graph of finite abelian groups
  68. Detectable sensation of a stochastic smoking model
  69. Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators
  70. Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms
  71. Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion
  72. On surrounding quasi-contractions on non-triangular metric spaces
  73. SEMT valuation and strength of subdivided star of K 1,4
  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
  75. Gradient estimates for a weighted nonlinear parabolic equation and applications
  76. On the equivalence of three-dimensional differential systems
  77. Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests
  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
  87. Towards a homological generalization of the direct summand theorem
  88. A standard form in (some) free fields: How to construct minimal linear representations
  89. On the determination of the number of positive and negative polynomial zeros and their isolation
  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
  91. Simply connected topological spaces of weighted composition operators
  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
  95. On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups
  96. Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow
  97. Noetherian properties in composite generalized power series rings
  98. Inequalities for the generalized trigonometric and hyperbolic functions
  99. Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions
  100. A new characterization of a proper type B semigroup
  101. Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields
  102. Estimates of entropy numbers in probabilistic setting
  103. Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
  104. S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems
  105. The logarithmic mean of two convex functionals
  106. A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation
  107. Approximation properties of tensor norms and operator ideals for Banach spaces
  108. A multi-power and multi-splitting inner-outer iteration for PageRank computation
  109. The edge-regular complete maps
  110. Ramanujan’s function k(τ)=r(τ)r2(2τ) and its modularity
  111. Finite groups with some weakly pronormal subgroups
  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
  114. The limit Riemann solutions to nonisentropic Chaplygin Euler equations
  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
  121. A study on strongly convex hyper S-subposets in hyper S-posets
  122. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
  123. Special Issue on Graph Theory (GWGT 2019), Part II
  124. On applications of bipartite graph associated with algebraic structures
  125. Further new results on strong resolving partitions for graphs
  126. The second out-neighborhood for local tournaments
  127. On the N-spectrum of oriented graphs
  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
  132. On triangular matroids induced by n3-configurations
  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
https://doi.org/10.1515/math-2020-0006
Keywords for this article
extremal graphs; tricyclic graphs; F-index
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  69. Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators
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  71. Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion
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  73. SEMT valuation and strength of subdivided star of K 1,4
  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
  75. Gradient estimates for a weighted nonlinear parabolic equation and applications
  76. On the equivalence of three-dimensional differential systems
  77. Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests
  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
  87. Towards a homological generalization of the direct summand theorem
  88. A standard form in (some) free fields: How to construct minimal linear representations
  89. On the determination of the number of positive and negative polynomial zeros and their isolation
  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
  91. Simply connected topological spaces of weighted composition operators
  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
  95. On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups
  96. Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow
  97. Noetherian properties in composite generalized power series rings
  98. Inequalities for the generalized trigonometric and hyperbolic functions
  99. Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions
  100. A new characterization of a proper type B semigroup
  101. Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields
  102. Estimates of entropy numbers in probabilistic setting
  103. Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
  104. S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems
  105. The logarithmic mean of two convex functionals
  106. A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation
  107. Approximation properties of tensor norms and operator ideals for Banach spaces
  108. A multi-power and multi-splitting inner-outer iteration for PageRank computation
  109. The edge-regular complete maps
  110. Ramanujan’s function k(τ)=r(τ)r2(2τ) and its modularity
  111. Finite groups with some weakly pronormal subgroups
  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
  114. The limit Riemann solutions to nonisentropic Chaplygin Euler equations
  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
  121. A study on strongly convex hyper S-subposets in hyper S-posets
  122. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
  123. Special Issue on Graph Theory (GWGT 2019), Part II
  124. On applications of bipartite graph associated with algebraic structures
  125. Further new results on strong resolving partitions for graphs
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  127. On the N-spectrum of oriented graphs
  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
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  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
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