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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
MSC
1 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
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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