R E S E A R C H
Open Access
On the general sum-connectivity index
and general Randi´c index of cacti
Shehnaz Akhter
1, Muhammad Imran
1,2*and Zahid Raza
3*Correspondence:
imrandhab@gmail.com 1School of Natural Sciences (SNS),
National University of Sciences and Technology (NUST), Sector H-12, Islamabad, Pakistan
2Department of Mathematical
Sciences, United Arab Emirates University, P.O. Box 15551, Al Ain, United Arab Emirates
Full list of author information is available at the end of the article
Abstract
LetGbe a connected graph. The degree of a vertexxofG, denoted bydG(x), is the number of edges adjacent tox. The general sum-connectivity index is the sum of the weights (dG(x) +dG(y))αfor all edgesxyofG, where
α
is a real number. The general Randi´c index is the sum of weights of (dG(x)dG(y))αfor all edgesxyofG, whereα
is a real number. The graphGis a cactus if each block ofGis either a cycle or an edge. In this paper, we find sharp lower bounds on the general sum-connectivity index and general Randi´c index of cacti.MSC: 05C90
Keywords: cacti; general sum-connectivity index; general Randi´c index
1 Introduction
LetGbe a finite molecular graph of ordernand sizemwith vertex setV(G) and edge set E(G). The degree of a vertexx∈V(G), denoted bydG(x), is the number of edges adjacent tox. A vertex with degree is called a pendent vertex. The minimum and maximum de-grees ofGare respectively defined byG=max{dG(x) :x∈V(G)}andδG=min{dG(x) : x∈V(G)}. The set of neighboring vertices of a vertexxis denoted byNG(x). The graph obtained by deleting a vertexx∈V(G) is denoted byG–x. The graph obtained fromGby adding an edgexybetween two nonadjacent verticesx,y∈V(G) is denoted byG+xy.
Historically, the first degree based topological indices are the Zagreb indices, introduced by Gutman and Trinajestić []. The first Zagreb indexM(G) and the second Zagreb index M(G) are defined as
M(G) =
x∈V(G)
dG(x)
and M(G) =
xy∈E(G)
dG(x)dG(y).
The general Randić index (or product-connectivity index) was proposed by Bollobás and Erdős [] and is defined as follows:
Rα(G) =
xy∈E(G)
dG(x)dG(y)
α
,
whereα is a real number. ThenR–/(G) =
xy∈E(G)(dG(x)dG(y))–
is called the Randić index, which was defined by Randić [], andRis the second Zagreb index. The Randić index is the most studied, most applied, and most important index among all
cal indices. Recently, Zhou and Trinajstić [] modified the concept of Randić index and obtained a new index, called the general sum-connectivity index and defined as follows:
χα(G) =
xy∈E(G)
dG(x) +dG(y)
α
,
whereαis a real number. Thenχ–/(G) =xy∈E(G)(dG(x) +dG(y))– is the sum-connec-tivity index defined by Zhou and Trinajstić [], and if α = , then the general sum-connectivity index becomes the first Zagreb index.
A connected graphGis a cactus if each block ofGis either a cycle or an edge. LetX(n,t) be the set of cacti of ordernwithtcycles. Obviously,X(n, ) is the set of trees of ordern, andX(n, ) is the set of unicyclic graphs of ordern. Denote byX(n,t) then-vertex cactus consisting ofttriangles andn– t– pendent edges such that triangles and pendent edges have exactly one vertex in common.
Lin et al. [] discuss the sharp lower bounds of the Randić index of cacti withrpendent vertices. Dong and Wu [] calculated the sharp bounds on the atom-bond connectivity index in the set of cacti withtcycles and also in the set of cacti withrpendent vertices, for all positive integral values oftandr. Li [] determined the unique cactus with maximum atom-bond connectivity index among cacti withnvertices and tcycles, where ≤t≤
n–
, and also among nvertices and rpendent vertices, where ≤r≤n– . Ma and Deng [] calculated the sharp lower bounds of the sum-connectivity index in the set of cacti of ordernwithtcycles and also in the set of cacti of order nwith perfect matching. Lu et al. [] gave the sharp lower bound on the Randić index of cacti withtnumber of cycles.
The present paper is motivated by the results of papers [, ]. 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
In this section, we find sharp lower bound for the general sum-connectivity index of cacti. Let
F(n,t) = t(n+ )α+ (n– t– )nα+ αt,
wheren≥ andt≥. First, we give some lemmas that will be used in the main result.
Lemma .([]) Let n≥be a positive integer,α< be a real number,and G∈X(n, ). Then
χα(G)≥(n– )nα.
Equality holds if and only if G∼=X(n, ).
Lemma .([]) Let n≥be a positive integer, –≤α< be a real number,and G∈ X(n, ).Then
χα(G)≥(n+ )α+ (n– )nα+ α.
Lemma . Let x,k be positive integers with x≥k≥,andα< be a real number.Define
f(x) =xα– (x–k)α.
Then f(x)is an increasing function.
Proof It is easily seen thatf (x) =α(xα–– (x–k)α–) > forα< . Therefore,f(x) is an
increasing function.
Lemma . Let x,k be positive integers with x≥k≥,and–≤α< be a real number. Define
f(x) =k(x+ )α+ (x–k)(x+ )α– (x–k)(x+ –k)α.
Then f(x)is a decreasing function.
Proof Letg(x) =k(x+ )α–k(x+ )αforx≥k≥. We getg(x) =αk((x+ )α–– (x+ )α–) < . So,g(x) is a decreasing function.
Leth(x) =x(x+ )αforx≥k≥. Thenh (x) =α((α+ )x+ )(x+ )α–< for –≤α< . Hence,h(x) –h(x–k) is a decreasing function. Note thatf(x) =g(x) + (h(x) –h(x–k)). Thus,
f(x) is a decreasing function.
Lemma . Let x,k be positive integers with x≥k≥,and–≤α< be a real number. Define
f(x) =x(x+ )α– (x– )xα.
Then f(x)is a decreasing function.
Proof Leth(x) =x(x+ )α forx≥k≥. Then h (x) =α((α+ )x+ )(x+ )α–< for –≤α< . Butf(x) =h(x) –h(x– ). Thus,f(x) is a decreasing function.
Theorem . Let–≤α< be a real number,n≥be a positive integer,and G∈X(n,t). Then
χα(G)≥F(n,t). (.)
Equality holds if and only if G∼=X(n,t).
Proof We use mathematical induction onnandt. Ift= ort= , then Lemmas . and . give inequality (.). Ifn= andt= , then there is only one possibility thatX(, ) is a graph with two cycles that have only one common vertex (see Figure ) and inequality (.) holds. Ifn≥ andt≥, then we will discuss the following two cases.
Case:G∈X(n,t) has at least one pendent vertex.
Figure 1 X0(5, 2).
are nonpendent vertices, wherek≥. If k= , then u,u,u, . . . ,ud– are nonpendent vertices. Obviously, we have t≤ n–k– . If H is a graph obtained by deleting vertices x,u,u, . . . ,uk–from G, that is,H=G–x–u–u–· · ·–uk–, thenH∈X(n–k,t). By inductive assumption and using Lemmas . and ., we obtain
χα(G) =χα(H) +k(d+ )α+
d–
i=k
d+dG(ui)α–d–k+dG(ui)α
≥F(n–k,t) +k(d+ )α+
d–
i=k
d+dG(ui)
α
–d–k+dG(ui)
α
=F(n,t) + t(n–k+ )α– t(n+ )α+ (n–k– t– )(n–k)α– (n– t– )nα
+k(d+ )α+
d–
i=k
d+dG(ui)
α
–d–k+dG(ui)
α
≥F(n,t) + t(n–k+ )α– t(n+ )α+ (n–k– t– )(n–k)α– (n– t– )nα
+k(d+ )α+ (d–k)(d+ )α– (d–k+ )α
≥F(n,t) + t(n–k+ )α– t(n+ )α+ (n–k– t– )(n–k)α– (n– t– )nα
+knα+ (n– –k)(n+ )α– (n– –k)(n+ –k)α
=F(n,t) + (n– –k– t)(n+ )α– (n–k+ )α–nα+ (n–k)α
≥F(n,t).
The equality holds if and only ifd=n– , t=n–k– , andH∼=X(n–k,t). Therefore, we have thatχα(G) =F(n,t) if and only ifG∼=X(n,t).
Case :G∈X(n,t) has no pendent vertex. Then we consider edgesxx,xx∈E(G) such thatxandxhave degree andxhas degreed, whered≥. Now there arise two subcases.
Subcase.:xandxare nonadjacent vertices inG. Then we construct a new graphH by deleting a vertexxand adding an edge betweenxandx, that is,H=G–x+xx∈ X(n– ,t). Then by inductive assumption we have
χα(G) =χα(H) + (d+ )α+ α– (d+ )α
≥F(n– ,t) + α
=F(n,t) + tnα– t(n+ )α+ (n– t– )(n– )α– (n– t– )nα+ α
=F(n,t) + tnα– (n– )α+ (n– t– )(n– )α–nα+ α–nα
>F(n,t).
Subcase.:xandxare adjacent inG. Then we construct a new graphHby deleting two verticesxandx, that is,H=G–x–x∈X(n– ,t– ). Let the set of neighbors ofx beNG(x)\ {x,x}={u,u, . . . ,ud–}, where allu,u, . . . ,ud– are nonpendent vertices. By inductive assumption and using Lemmas . and ., we obtain
χα(G) =χα(H) + α+ (d+ )α+
d–
i=
d+dG(ui)α–d– +dG(ui)α
≥F(n– ,t– ) + α+ (d+ )α+ d–
i=
d+dG(ui)α–d– +dG(ui)
α
=F(n,t) + (t– )(n– )α– t(n+ )α+ (n– t– )(n– )α– (n– t– )nα
+ (t– )α–tα+ α+ (d+ )α+ d–
i=
d+dG(ui)α–d– +dG(ui)
α
=F(n,t) + (t– )(n– )α– t(n+ )α+ (n– t– )(n– )α– (n– t– )nα
+ (t– )α–tα+ α+ (d+ )α+ (d– )(d+ )α–dα
=F(n,t) + (t– )(n– )α– t(n+ )α+ (n– t– )(n– )α– (n– t– )nα
+d(d+ )α– (d– )dα
=F(n,t) + (t– )(n– )α– t(n+ )α+ (n– t– )(n– )α– (n– t– )nα
+ (n– )(n+ )α– (n– )(n– )α
=F(n,t) + (n– – t)(n+ )α– (n– )α+ (n– )α–nα
≥F(n,t).
The equality holds if and only ifd=n– , t=n– , andH∼=X(n– ,t– ). Therefore, we haveχα(G) =F(n,t) with equality if and only ifG∼=X(n,t).
3 General Randi´c index
In this section, we find sharp lower bound for the general Randić index of cacti. Let
F(n,t) = α+t(n– ) +tα+ (n– – t)(n– )α,
wheren≥ andt≥. First, we give some lemmas that will be helpful in the proof of the main result.
Lemma .([]) Let n≥be a positive integer,α< be a real number,and G∈X(n, ). Then
Rα(G)≥(n– )α+.
Equality holds if and only if G∼=X(n, ).
Rα(G)≥(n– )(n– )α+ α(n– )α+ α.
Equality holds if and only if G∼=X(n, ).
Lemma . Let x,d,k be positive integers with d≥k≥,x≥,andα< be a real number. Define
f(x) =xαdα– (d–k)α.
Then f(x)is an increasing function.
Proof It is easily seen thatf (x) =αxα–(dα– (d–k)α) > forα< . Therefore,f(x) is an
increasing function.
Lemma . Let x,k be positive integers with x≥k≥,and–≤α< be a real number. Define
f(x) =kxα+ α(x–k)xα– α(x–k)α+.
Then f(x)is a decreasing function.
Proof Letg(x) =kxα( – α) forx≥k≥. We getg(x) =αkxα–( – α) < . So,g(x) is a
decreasing function.
Leth(x) = αxα+ forx≥k≥. Thenh (x) = α(α+ )xα–< for –≤α< . Hence, h(x) –h(x–k) is a decreasing function. Note thatf(x) =g(x) + (h(x) –h(x–k)). Thus,f(x)
is a decreasing function.
Lemma . Let x,k be positive integers with x≥k≥,and–≤α< be areal number. Let
f(x) =xα+– (x– )α+.
Then f(x)is a decreasing function.
Proof Leth(x) =xα+ forx≥k≥. Then h (x) =α(α+ )xα–< for –≤α< . But f(x) =h(x) –h(x– ). Thusf(x) is a decreasing function.
Theorem . Let–≤α< be a real number,n≥be a positive integer,and G∈X(n,t). Then
Rα(G)≥F(n,t). (.)
Equality holds if and only if G∼=X(n,t).
Case:G∈X(n,t) has at least one pendent vertex.
Letxbe a vertex that is adjacent only with vertexy. Thenyhas degreedwith ≤d≤ n– . The set of neighbors ofyinGisNG(y)\{x}={u,u,u, . . . ,ud–}. Without lost of gen-erality, we assume thatdG(uj)≥ fork≤j≤d– , wherek≥, and the remaining neigh-boring vertices ofyare pendent vertices. Ifk= , thenGhas no pendent vertex. Obviously, we havet≤ n–k– . IfHis a new graph obtained by deleting verticesx,u,u, . . . ,uk–, that is,H=G–x–u–u–· · ·–uk–, thenH∈X(n–k,t). By inductive assumption and using Lemmas . and ., we obtain
Rα(G) =Rα(H) +kdα+
d–
j=k
ddG(uj)α– (d–k)αdG(uj)α
≥F(n–k,t) +kdα+ d–
j=k
ddG(uj)
α
– (d–k)αd
G(uj)α
=F(n,t) + α+t(n–k+ )α– α+t(n– )α+ (n–k– t– )(n–k– )α
– (n– t– )(n– )α+kdα+ d–
j=k
ddG(uj)
α
– (d–k)αdG(uj)α
≥F(n,t) + α+t(n–k+ )α– α+t(n– )α+ (n–k– t– )(n–k– )α
– (n– t– )(n– )α+kdα+ (d–k)(d)α– α(d–k)α
≥F(n,t) + α+t(n–k+ )α– α+t(n– )α+ (n–k– t– )(n–k– )α
– (n– t– )(n– )α+k(n– )α+ α(n– –k)(n– )α– α(n+ –k)α+
=F(n,t) + (n– –k– t)
– α(n–k– )α– (n– )α
≥F(n,t).
The equality holds if and only ifH∼=X(n–k,t),d=n– , and t=n–k– . Therefore, we have thatRα(G) =F(n,t) if and only ifG∼=X(n,t).
Case: IfGhas no pendent vertex, then we considerxx,xxedges of a cycle such that xandxare vertices of degree two anddG(x) =d, whered≥. Next, we discuss this in the following two subcases.
Subcase.: Ifxandxare nonadjacent vertices inG, thenHis a new graph obtained by deletingxand adding an edgexx, that is,H=G–x+xx∈X(n– ,t). Therefore, we get
Rα(G) =Rα(H) + (d)α+ α– (d)α
≥F(n– ,t) + α
=F(n,t) + α+t(n– )α– α+t(n– )α+ (n– t– )(n– )α
– (n– t– )(n– )α+ α
=F(n,t) + α+t
(n– )α– (n– )α+ (n– t– )(n– )α–nα
+ α– (n– )α
The last inequality holds ifn≥. Ifn= , thent≤, and we easily show the inequal-ity.
Subcase.: Ifxandxare connected by an edge, thenH=G–x–x∈X(n– ,t– ). Let the set of neighbors ofxbeNG(x)\ {x,x}={u,u, . . . ,ud–}, whereujfor ≤j≤ d– are nonpendent vertices. By inductive assumption and using Lemmas . and ., we obtain
Rα(G) =Rα(H) + α+ α+dα+
d–
j=
ddG(uj)
α
– (d– )αd
G(uj)α
≥F(n– ,t– ) + α+ α+dα+ d–
j=
ddG(uj)α– (d– )αdG(uj)α
=F(n,t) + α+(t– )(n– )α– α+t(n– )α+ (n– t– )(n– )α
– (n– t– )(n– )α+ (t– )α–tα+ α+ α+dα
+ d–
j=
ddG(uj)
α
– (d– )α+dG(uj)α
=F(n,t)α+(t– )(n– )α– α+t(n– )α+ (n– t– )(n– )α
– (n– t– )(n– )α+ (t– )α–tα+ α+ α+dα
+ α(d– )dα– (d– )α
=F(n,t) + α+(t– )(n– )α– α+t(n– )α+ (n– t– )(n– )α
– (n– t– )(n– )α+ αdα+– α(d– )α+
=F(n,t) + α+(t– )(n– )α– α+t(n– )α+ (n– t– )(n– )α
– (n– t– )(n– )α+ α(n– )α– α(n– )α
=F(n,t) + (n– – t) – α(n– )α– (n– )α
≥F(n,t).
The equality holds if and only ifH∼=X(n– ,t– ),d=n– , and t=n– . Therefore, we have thatRα(G) =F(n,t) if and only ifG∼=X(n,t).
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 –≤α< . The general sum-connectivity index and general Randić index of cacti for other values of
αremains 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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
was prepared by Shehnaz Akhter, which was verified and improved by Muhammad Imran and Zahid Raza. The final version was prepared by Shenaz Akhter and Muhammad Imran. All authors read and approved the final manuscript.
Author details
1School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Sector H-12, Islamabad, Pakistan. 2Department of Mathematical Sciences, United Arab Emirates University, P.O. Box 15551, Al Ain, United Arab Emirates.3Department of Mathematics, College of Sciences, University of Sharjah, Sharjah, United Arab Emirates.
Received: 24 September 2016 Accepted: 16 November 2016
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