On the general sum-connectivity index and general Randić index of cacti
- Shehnaz Akhter^{1},
- Muhammad Imran^{1, 2}Email author and
- Zahid Raza^{3}
https://doi.org/10.1186/s13660-016-1250-6
© The Author(s) 2016
Received: 24 September 2016
Accepted: 16 November 2016
Published: 22 November 2016
Abstract
Let G be a connected graph. The degree of a vertex x of G, denoted by \(d_{G}(x)\), is the number of edges adjacent to x. The general sum-connectivity index is the sum of the weights \((d_{G}(x)+d_{G}(y))^{\alpha}\) for all edges xy of G, where α is a real number. The general Randić index is the sum of weights of \((d_{G}(x)d_{G}(y))^{\alpha}\) for all edges xy of G, where α is a real number. The graph G is a cactus if each block of G is either a cycle or an edge. In this paper, we find sharp lower bounds on the general sum-connectivity index and general Randić index of cacti.
Keywords
cacti general sum-connectivity index general Randić indexMSC
05C901 Introduction
Let G be a finite molecular graph of order n and size m with vertex set \(V(G)\) and edge set \(E(G)\). The degree of a vertex \(x\in V(G)\), denoted by \(d_{G}(x)\), is the number of edges adjacent to x. A vertex with degree 1 is called a pendent vertex. The minimum and maximum degrees of G are respectively defined by \(\bigtriangleup _{G}=\max\{d_{G}(x): x\in V(G)\}\) and \(\delta_{G}=\min\{d_{G}(x):x\in V(G)\}\). The set of neighboring vertices of a vertex x is denoted by \(N_{G}(x)\). The graph obtained by deleting a vertex \(x\in V(G)\) is denoted by \(G-x\). The graph obtained from G by adding an edge xy between two nonadjacent vertices \(x,y\in V(G)\) is denoted by \(G+xy\).
A connected graph G is a cactus if each block of G is either a cycle or an edge. Let \(X(n,t)\) be the set of cacti of order n with t cycles. Obviously, \(X(n,0)\) is the set of trees of order n, and \(X(n,1)\) is the set of unicyclic graphs of order n. Denote by \(X^{0}(n,t)\) the n-vertex cactus consisting of t triangles and \(n-2t-1\) pendent edges such that triangles and pendent edges have exactly one vertex in common.
Lin et al. [6] discuss the sharp lower bounds of the Randić index of cacti with r pendent vertices. Dong and Wu [7] calculated the sharp bounds on the atom-bond connectivity index in the set of cacti with t cycles and also in the set of cacti with r pendent vertices, for all positive integral values of t and r. Li [8] determined the unique cactus with maximum atom-bond connectivity index among cacti with n vertices and t cycles, where \(0\leq t\leq\lfloor\frac{n-1}{2}\rfloor\), and also among n vertices and r pendent vertices, where \(0\leq r\leq n-1\). Ma and Deng [9] calculated the sharp lower bounds of the sum-connectivity index in the set of cacti of order n with t cycles and also in the set of cacti of order 2n with perfect matching. Lu et al. [10] gave the sharp lower bound on the Randić index of cacti with t number of cycles.
The present paper is motivated by the results of papers [9, 10]. The goal of this paper is to compute the sharp lower bounds for the general sum-connectivity index and the general Randić index of cacti with fixed number of cycles.
2 General sum-connectivity index
Lemma 2.1
[4]
Lemma 2.2
[11]
Lemma 2.3
Proof
It is easily seen that \(f'(x)=\alpha(x^{\alpha-1}-(x-k)^{\alpha-1})>0\) for \(\alpha<0\). Therefore, \(f(x)\) is an increasing function. □
Lemma 2.4
Proof
Let \(g(x)=k(x+1)^{\alpha}-k(x+2)^{\alpha}\) for \(x\geq k\geq1\). We get \(g'(x)={\alpha}k ((x+1)^{\alpha-1}-(x+2)^{\alpha-1} )<0\). So, \(g(x)\) is a decreasing function.
Let \(h(x)=x(x+2)^{\alpha}\) for \(x\geq k\geq1\). Then \(h''(x)=\alpha((\alpha +1)x+4)(x+2)^{\alpha-2}<0\) for \(-1\leq\alpha<0\). Hence, \(h(x)-h(x-k)\) is a decreasing function. Note that \(f(x)=g(x)+(h(x)-h(x-k))\). Thus, \(f(x)\) is a decreasing function. □
Lemma 2.5
Proof
Let \(h(x)=x(x+2)^{\alpha}\) for \(x\geq k\geq1\). Then \(h''(x)=\alpha((\alpha +1)x+4)(x+2)^{\alpha-2}<0\) for \(-1\leq\alpha<0\). But \(f(x)=h(x)-h(x-2)\). Thus, \(f(x)\) is a decreasing function. □
Theorem 2.1
Proof
Case 1: \(G\in X(n,t)\) has at least one pendent vertex.
The equality holds if and only if \(d=n-1\), \(2t=n-k-1\), and \(H\cong X^{0}(n-k,t)\). Therefore, we have that \(\chi_{\alpha}(G)=F(n,t)\) if and only if \(G\cong X^{0}(n,t)\).
Case 2: \(G\in X(n,t)\) has no pendent vertex. Then we consider edges \(x_{0}x_{1}, x_{0}x_{2}\in E(G)\) such that \(x_{0}\) and \(x_{1}\) have degree 2 and \(x_{2}\) has degree d, where \(d\geq3\). Now there arise two subcases.
3 General Randić index
Lemma 3.1
[12]
Lemma 3.2
[13]
Lemma 3.3
Proof
It is easily seen that \(f'(x)=\alpha x^{\alpha-1}(d^{\alpha }-(d-k)^{\alpha})>0\) for \(\alpha<0\). Therefore, \(f(x)\) is an increasing function. □
Lemma 3.4
Proof
Let \(g(x)=kx^{\alpha}(1-2^{\alpha})\) for \(x\geq k\geq1\). We get \(g'(x)={\alpha}kx^{\alpha-1}(1-2^{\alpha})<0\). So, \(g(x)\) is a decreasing function.
Let \(h(x)=2^{\alpha}x^{\alpha+1}\) for \(x\geq k\geq1\). Then \(h''(x)=2^{\alpha}(\alpha+1)x^{\alpha-1}<0\) for \(-1\leq\alpha<0\). Hence, \(h(x)-h(x-k)\) is a decreasing function. Note that \(f(x)=g(x)+(h(x)-h(x-k))\). Thus, \(f(x)\) is a decreasing function. □
Lemma 3.5
Proof
Let \(h(x)=x^{\alpha+1}\) for \(x\geq k\geq1\). Then \(h''(x)=\alpha(\alpha +1)x^{\alpha-1}<0\) for \(-1\leq\alpha<0\). But \(f(x)=h(x)-h(x-2)\). Thus \(f(x)\) is a decreasing function. □
Theorem 3.1
Proof
We use mathematical induction on n and t. If \(t=0\) or \(t=1\), then inequality (3.1) holds by Lemma 3.1 and Lemma 3.2, respectively. If \(n=5\) and \(t=2\), then there is only one graph \(X^{0}(5,2)\), and the result for \(X^{0}(5,2)\) is trivial. Now if \(n\geq6\) and \(t\geq2\), then we will consider the following two cases.
Case 1: \(G\in X(n,t)\) has at least one pendent vertex.
The equality holds if and only if \(H\cong X^{0}(n-k,t)\), \(d=n-1\), and \(2t=n-k-1\). Therefore, we have that \(R_{\alpha}(G)=F_{1}(n,t)\) if and only if \(G\cong X^{0}(n,t)\).
Case 2: If G has no pendent vertex, then we consider \(x_{0}x_{1},x_{0}x_{2}\) edges of a cycle such that \(x_{0}\) and \(x_{1}\) are vertices of degree two and \(d_{G}(x_{2})=d\), where \(d\geq3\). Next, we discuss this in the following two subcases.
4 Conclusion
In this paper, we determined the sharp lower bounds for the general sum-connectivity index and the general Randić index of cacti with fixed number of cycles for \(-1\leq\alpha<0\). The general sum-connectivity index and general Randić index of cacti for other values of α remains an open problem. Moreover, some topological indices and polynomials are still unknown for cacti with fixed number of cycles and fixed number of pendent vertices.
Declarations
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Authors’ Affiliations
References
- Gutman, I, Trinajstic, N: Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons. Chem. Phys. Lett. 17, 535-538 (1972) View ArticleGoogle Scholar
- Bollobás, B, Erdős, P: Graphs of extremal weights. Ars Comb. 50, 225-233 (1998) MathSciNetMATHGoogle Scholar
- Randić, M: On characterization of molecular branching. J. Am. Chem. Soc. 97, 6609-6615 (1975) View ArticleGoogle Scholar
- Zhou, B, Trinajstić, N: On general sum-connectivity index. J. Math. Chem. 47, 210-218 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Zhou, B, Trinajstić, N: On a novel connectivity index. J. Math. Chem. 46, 1252-1270 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Lin, A, Luo, R, Zha, X: A sharp lower of the Randić index of cacti with r pendants. Discrete Appl. Math. 156, 1725-1735 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Dong, H, Wu, X: On the atom-bond connectivity index of cacti. Filomat 28, 1711-1717 (2014) MathSciNetView ArticleGoogle Scholar
- Li, J: On the ABC index of cacti. Int. J. Graph Theory Appl. 1, 57-66 (2015) Google Scholar
- Ma, F, Deng, H: On the sum-connectivity index of cacti. Math. Comput. Model. 54, 497-507 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Lu, M, Zhang, L, Tian, F: On the Randić index of cacti. MATCH Commun. Math. Comput. Chem. 56, 551-556 (2006) MathSciNetMATHGoogle Scholar
- Du, Z, Zhou, B, Trinajstić, N: Minimum general sum-connectivity index of unicyclic graphs. J. Math. Chem. 48, 697-703 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Hu, Y, Li, X, Yuan, Y: Trees with minimum general Randić index. MATCH Commun. Math. Comput. Chem. 52, 119-128 (2004) MathSciNetMATHGoogle Scholar
- Wu, B, Zhang, L: Unicyclic graphs with minimum general Randić index. MATCH Commun. Math. Comput. Chem. 54, 455-464 (2005) MathSciNetMATHGoogle Scholar