Extremal Unicyclic Graphs Research Articles

Bounds on the edge version of the Nirmala index

The edge version of the Nirmala index is defined in terms of edge degrees as [Formula: see text], where [Formula: see text] denotes the degree of the edge [Formula: see text] in graph [Formula: see text] and [Formula: see text] means that the edges [Formula: see text] and [Formula: see text] are adjacent. In this paper, we obtain some sharp bounds for the edge version of the Nirmala index in terms of the edge version of topological indices. Also, we give the extremal unicyclic graphs of order [Formula: see text] with the minimum edge version of the Nirmala index.

General Randić Index of Unicyclic Graphs and Its Applications to Drugs

In this work, we determine the maximum general Randić index (a general symmetric function of vertex degrees) for η0≤η<0 among all n-vertex unicyclic graphs with a fixed maximum degree Δ and the maximum and the second maximum general Randić index for η0≤η<0 among all n-vertex unicyclic graphs, where η0≈−0.21. We establish sharp inequalities and identify the graphs attaining the inequalities. Thereby, extremal graphs are obtained for the general Randić index, and certain open gaps in the theory of extremal unicyclic graphs are filled (some open problems are provided). We use computational software to calculate the Randić index for the chemical trees up to order 7 and use the statistical (linear regression) analysis to discuss the various applications of the Randić index with the physical properties of drugs on the said chemical trees. We show that the Randić index is better correlated with the heat of vaporization for these alkanes.

Extremal unicyclic graphs of Sombor index

Unicyclic graph is one kind of typical graph for indices basing degree of vertex. Various chemical structures can be exhibited in unicyclic graph as well. A graph is said to be unicyclic if the graph is connected and |V(G)|=|E(G)|. Recently, the Sombor index, which is defined bySO=SO(G)=∑uv∈E(G)dG2(u)+dG2(v), was proposed by Gutman. This paper establishes distinct bounds for this index of unicyclic graph with girth l, as well as specific bounds of chemical unicyclic graph with girth l.

Upper and lower bounds for topological indices on unicyclic graphs

The aim of this paper is to obtain new inequalities for a large family of topological indices restricted to unicyclic graphs and to characterize the set of extremal unicyclic graphs with respect to them. This family includes variable first Zagreb, variable sum exdeg, multiplicative second Zagreb and Narumi-Katayama indices. Our main results provide upper and lower bounds for these topological indices on unicyclic graphs, fixing or not the maximum degree or the number of pendant vertices.

The extremal unicyclic graphs with given diameter and minimum edge revised Szeged index

<abstract><p>Let $ H $ be a connected graph. The edge revised Szeged index of $ H $ is defined as $ Sz^{\ast}_{e}(H) = \sum\limits_{e = uv\in E_H}(m_{u}(e|H)+\frac{m_{0}(e|H)}{2})(m_{v}(e|H)+\frac{m_{0}(e|H)}{2}) $, where $ m_{u}(e|H) $ (resp., $ m_{v}(e|H) $) is the number of edges whose distance to vertex $ u $ (resp., $ v $) is smaller than to vertex $ v $ (resp., $ u $), and $ m_{0}(e|H) $ is the number of edges equidistant from $ u $ and $ v $. In this paper, the extremal unicyclic graphs with given diameter and minimum edge revised Szeged index are characterized.</p></abstract>

Characterization of Extremal Unicyclic Graphs with Fixed Leaves Using the Lanzhou Index

A topological index being a graph theoretic parameter plays a role of function for the assignment of a numerical value to a molecular graph which predicts the several physical and chemical properties of the underlying molecular graph such as heat of evaporation, critical temperature, surface tension, boiling point, octanol-water partition coefficient, density and flash points. For a (molecular) graph Γ, the Lanzhou index (Lz index) is obtained by the sum of deg(v)2de¯g(v) over all the vertices, where deg(v) and de¯g(v) are degrees of the vertex v in Γ and its complement Γ¯ respectively. Let Vαβ be a class of unicyclic graphs (same order and size) such that each graph of this class has order α and β leaves (vertices of degree one). In this note, we compute the lower and upper bounds of Lz index for each unicyclic graph in the class of graphs Vαβ. Moreover, we characterize the extremal graphs with respect to Lz index in the same class of graphs.

On General Reduced Second Zagreb Index of Graphs

Graph-based molecular structure descriptors (often called “topological indices”) are useful for modeling the physical and chemical properties of molecules, designing pharmacologically active compounds, detecting environmentally hazardous substances, etc. The graph invariant GRMα, known under the name general reduced second Zagreb index, is defined as GRMα(Γ)=∑uv∈E(Γ)(dΓ(u)+α)(dΓ(v)+α), where dΓ(v) is the degree of the vertex v of the graph Γ and α is any real number. In this paper, among all trees of order n, and all unicyclic graphs of order n with girth g, we characterize the extremal graphs with respect to GRMα(α≥−12). Using the extremal unicyclic graphs, we obtain a lower bound on GRMα(Γ) of graphs in terms of order n with k cut edges, and completely determine the corresponding extremal graphs. Moreover, we obtain several upper bounds on GRMα of different classes of graphs in terms of order n, size m, independence number γ, chromatic number k, etc. In particular, we present an upper bound on GRMα of connected triangle-free graph of order n>2, m>0 edges with α>−1.5, and characterize the extremal graphs. Finally, we prove that the Turán graph Tn(k) gives the maximum GRMα(α≥−1) among all graphs of order n with chromatic number k.

Extremal Unicyclic Graphs with Respect to Vertex-Degree-Based Topological Indices

n this paper we present a general criteria to decide when the cycle on vertices and , the coalescence of the star with the cycle , are extremal unicyclic graphs of a vertex-degree-based (VDB) topological index. We show that many of the well known results on extremal values of VDB topological indices over unicyclic graphs can be obtained as particular cases of ours. Moreover, we obtain new results on extremal values of VDB topological indices, such as the generalized Geometric-Arithmetic indices, the generalized Atom-Bond-Connectivity indices, and its exponentials, among others.

Characterization of Extremal Unicyclic Graphs Using F-Coindex

The study of forgotten index and coindex for the molecular structures of some special chemical graphs (compounds and drugs) has proven significant in medical and pharmaceutical drug design fields by making reliable statistical conclusion about biological properties of new chemical compounds and drugs. In mathematical chemistry, finding extreme graphs with respect to topological index is an active area of research. The aim of this paper is to characterize the family of unicyclic graphs with extreme (largest and smallest) F-coindex. Moreover, the study also contains different other properties of unicyclic graphs. All these results are based upon an alternative form of F-coindex which is established with the help of special property in the graphs.

Extremal unicyclic graphs with respect to the Sanskruti index

Sanskruti index of a graph [Formula: see text] with vertex-set [Formula: see text] and edge-set [Formula: see text] is [Formula: see text], where [Formula: see text]. In this paper, the extremal graphs in the class of unicyclic graphs are characterized with respect to the Sanskruti index, where the considered class of unicyclic graphs contains five different large families of graphs.

Graph entropy based on the number of spanning forests of c-cyclic graphs

Graph entropies have been introduced to quantitatively measure the structural information content of graphs and networks; they have plenty of applications in various fields. Utilizing the number of subgraphs to establish measures for determining the complexity of molecular graphs are also prevalent in the study of mathematical chemistry. In this paper, we develop a new graph entropy measure that is based on the number of spanning forests. We prove explicit expressions for the entropy for trees, unicyclic and bicyclic graphs, and show that the cycle graph Cn attains the maximal value of the entropy for unicyclic graphs with order n and large cycle lengths. Based on generating numerical results, we conjecture extremal unicyclic graphs with respect to the entropy as well as we compare the values of our entropy for c-cyclic graphs, and generate graphs of bicyclic graphs and tricyclic graphs with 6 vertices for performing further research.

On the permanental sum of graphs

Let G be a graph and A(G) the adjacency matrix of G. The polynomial π(G,x)=per(xI−A(G)) is called the permanental polynomial of G, and the permanental sum of G is the summation of the absolute values of the coefficients of π(G, x). In this paper, we investigate properties of permanental sum of a graph, prove recursive formulas to compute the permanental sum of a graph, and show that the ordering of graphs with respect to permanental sum. Furthermore, we determine the upper and lower bounds of permanental sum of unicyclic graphs, and the corresponding extremal unicyclic graphs are also determined.

The harmonic index of unicyclic graphs

The harmonic index of a graph [Formula: see text], denoted by [Formula: see text], is defined as the sum of weights [Formula: see text] over all edges [Formula: see text] of [Formula: see text], where [Formula: see text] denotes the degree of a vertex [Formula: see text]. Hu and Zhou [WSEAS Trans. Math. 12 (2013) 716–726] proved that for any unicyclic graph [Formula: see text] of order [Formula: see text], [Formula: see text] with equality if and only if [Formula: see text]. Recently, Zhong and Cui [Filomat 29 (2015) 673–686] generalized the above bound and proved that for any unicyclic graph [Formula: see text] of order [Formula: see text] other than [Formula: see text], [Formula: see text]. In this paper, we generalize the aforemention results and show that for any connected unicyclic graph [Formula: see text] of order [Formula: see text] with maximum degree [Formula: see text], [Formula: see text] and classify the extremal unicyclic graphs.

Extremal unicyclic graphs with respect to additively weighted Harary index

In this paper we define cycle-star graphCSk;n k to be a graph onn vertices consisting of the cycle of lengthk andn k leafs appended to the same vertex of the cycle. Also, we define cycle-path graphCPk;n k to be a graph onn vertices consisting of the cycle of lengthk and of path on n k vertices whose one end is linked to a vertex on a cycle. We establish that cycle- star graphCS3;n 3 is the only maximal graph with respect to additively weighted Harary index among all unicyclic graphs on n vertices, while cycle-path graph CP3;n 3 is the only minimal unicyclic graph (here n must be at least 5): The values of additively weighted Harary index for extremal unicyclic graphs are established, so these values are the upper and the lower bound for the value of additively weighted Harary index on the class of unicyclic graphs onn vertices. 2010 Mathematics Subject Classification: 05C35

Some results on chemical energy of graphs

The chemical energy of an n-vertex graph G is defined as CE(G)={2∑i=1n2λi(G)if n is even ,2∑i=1n−12λi(G)+λn+12(G)if n is odd . Here λ1(G),λ2(G),…,λn(G) are the eigenvalues of G arranged in non-increasing order. We give lower and upper bounds for the chemical energy of graphs, and determine the extremal unicyclic graphs with minimal and maximal chemical energies.

Signless Laplacian coefficients and incidence energy of unicyclic graphs with the matching number

Let be the characteristic polynomial of the signless Laplacian matrix of a simple graph of order , where and are the degree diagonal and adjacency matrices of , respectively. In this paper, we focus on how the signless Laplacian coefficients of unicyclic graphs change after some graph transformations. These results can be used to characterize all extremal unicyclic graphs having the minimal signless Laplacian coefficients in the set of all unicyclic graphs of order and the matching number . Moreover, the unicyclic graphs with minimum incidence energy in are also characterized.

Extremal unicyclic graphs with minimal distance spectral radius

Extremal unicyclic graphs with minimal distance spectral radius

The second Zagreb indices of unicyclic graphs with given degree sequences

Let π=(d1,d2,…,dn) and π′=(d1′,d2′,…,dn′) be two different non-increasing degree sequences. We write π◃π′, if and only if ∑i=1ndi=∑i=1ndi′, and ∑i=1jdi≤∑i=1jdi′ for all j=1,2,…,n. Let Γ(π) be the class of connected graphs with degree sequence π. The second Zagreb index of a graph G is denoted by M2(G)=∑uv∈E(G)d(u)d(v). In this paper, we characterize an extremal unicyclic graph that achieves the maximum second Zagreb index in the class of unicyclic graphs with given degree sequence, and we also prove that if π◃π′, π and π′ are unicyclic degree sequences and U∗ and U∗∗ have the maximum second Zagreb indices in Γ(π) and Γ(π′), respectively, then M2(U∗)<M2(U∗∗). Furthermore, we determine the first to ninth largest second Zagreb indices together with the corresponding extremal unicyclic graphs in the class of unicyclic graphs on n≥17 vertices.

On the Maximal Eccentric Distance Sums of Graphs

If G is a simple connected graph with vertex V(G), then the eccentric distance sum of G, denoted by ξd(G), is defined as ∑v∈V(G)ecG(v)DG(v), where ecG(v) is the eccentricity of the vertex v and DG(v) is the sum of all distances from the vertex v. Let n≥8. We determine the n-vertex trees with, respectively, the maximum, second-maximum, third-maximum, and fourth-maximum eccentric distance sums. We also characterize the extremal unicyclic graphs on n vertices with respectively, the maximal, second maximal, and third maximal eccentric distance sums.

On the eccentric distance sum of trees and unicyclic graphs

Let G be a simple connected graph with the vertex set V ( G ) . The eccentric distance sum of G is defined as ξ d ( G ) = ∑ v ∈ V ( G ) ε ( v ) D G ( v ) , where ε ( v ) is the eccentricity of the vertex v and D G ( v ) = ∑ u ∈ V ( G ) d ( u , v ) is the sum of all distances from the vertex v. In this paper we characterize the extremal unicyclic graphs among n-vertex unicyclic graphs with given girth having the minimal and second minimal eccentric distance sum. In addition, we characterize the extremal trees with given diameter and minimal eccentric distance sum.