A connected graph $G$ is said to be $k$-connected if it has more than $k$ vertices and remains connected whenever fewer than $k$ vertices are deleted. In this paper, for a connected graph $G$ with sufficiently large order, we present a tight sufficient condition for $G$ with fixed minimum degree to be $k$-connected based on the $Q$-index. Our result can be viewed as a spectral counterpart of the corresponding Dirac type condition.
The Harary index of a graph is defined as the sum of reciprocals of distances between all pairs of vertices of the graph. In this paper we provide an upper bound of the Harary index in terms of the vertex or edge connectivity of a graph. We characterize the unique graph with maximum Harary index among all graphs with given number of cut vertices or vertex connectivity or edge connectivity. In addition we also characterize the extremal graphs with the second maximum Harary index among the graphs with given vertex connectivity.
The eccentric connectivity index, proposed by Sharma, Goswami and Madan, has been employed successfully for the development of numerous mathematical models for the prediction of biological activities of diverse nature. We now report mathematical properties of the eccentric connectivity index. We establish various lower and upper bounds for the eccentric connectivity index in terms of other graph invariants including the number of vertices, the number of edges, the degree distance and the first Zagreb index. We determine the n-vertex trees of diameter with the minimum eccentric connectivity ind…
In this paper we establish all extremal graphs with respect to augmented eccentric connectivity index among all (simple connected) graphs, among trees and among trees with perfect matching. For graphs that turn out to be extremal explicit formulas for the value of augmented eccentric connectivity index are derived.
We determine the minimum sum--connectivity index of bicyclic graphs with $n$ vertices and matching number $m$, where $2\le m\le \lfloor\frac{n}{2}\rfloor$, the minimum and the second minimum, as well as the maximum and the second maximum sum--connectivity indices of bicyclic graphs with $n\ge 5$ vertices. The extremal graphs are characterized.
The general sum-connectivity index of a graph $G$ is defined as $χ_α(G)=\sum\limits_{uv\in E(G)} {(d(u)+d(v))^α}$, where $d(v)$ denotes the degree of the vertex $v$ in $G$ and $α$ is a real number. In this paper it is deduced the maximum value for the general sum-connectivity index of $n$-vertex trees for $-1.7036\leq α<0$ and of $n$-vertex unicyclic graphs for $-1\le α<0$ respectively, with fixed maximum degree $\triangle $. The corresponding extremal graphs, as well as the $n$-vertex unicyclic graphs with the second maximum general sum-connectivity index for $n\ge 4$ are characterized. This …
The atom-bond connectivity (ABC) index is a degree-based topological index. It was introduced due to its applications in modeling the properties of certain molecular structures and has been since extensively studied. In this note, we examine the influence on the extremal values of the ABC index by various graph parameters. More specifically, we consider the maximum ABC index of connected graphs of given order, with fixed independence number, number of pendent vertices, chromatic number and edge-connectivity respectively. We provide characterizations of extremal structures as well as some conje…
The eccentric connectivity index $ξ^c$ is a novel distance--based molecular structure descriptor that was recently used for mathematical modeling of biological activities of diverse nature. It is defined as $ξ^c (G) = \sum_{v \in V (G)} deg (v) \cdot ε(v)$\,, where $deg (v)$ and $ε(v)$ denote the vertex degree and eccentricity of $v$\,, respectively. We survey some mathematical properties of this index and furthermore support the use of eccentric connectivity index as topological structure descriptor. We present the extremal trees and unicyclic graphs with maximum and minimum eccentric connect…
The leap eccentric connectivity index of $G$ is defined as $$Lξ^{C}(G)=\sum_{v\in V(G)}d_{2}(v|G)e(v|G)$$ where $d_{2}(v|G) $ be the second degree of the vertex $v$ and $e(v|G)$ be the eccentricity of the vertex $v$ in $G$. In this paper, we first give a counterexample for that if $G$ be a graph and $S(G)$ be its the subdivision graph, then each vertex $v\in V(G)$, $e(v|S(G))=2e(v|G)$ by Yarahmadi in \cite{yar14} in Theorem 3.1. And we describe the upper and lower bounds of the leap eccentric connectivity index of four graphs based on subdivision edges, and then give the expressions of the lea…
The recently developed atom-bond sum-connection (ABS) index is a variation of the well-studied graph-based molecular descriptors connectivity (Randic), atom-bond connectivity, and sum-connectivity indices.We present the minimum ABS index of trees with a given number of pendent vertices in this paper, which provides a solution to the problem proposed by A. Ali, I. Gutman and I. Redzepovic, Atom-Bond Sum-Connectivity Index of Unicyclic Graphs and Some Applications, Electron. J. Math.(2023) 1-7
The eccentric connectivity index $ξ^c$ is a distance--based molecular structure descriptor that was recently used for mathematical modeling of biological activities of diverse nature. We prove that the broom has maximum $ξ^c$ among trees with a fixed maximum vertex degree, and characterize such trees with minimum $ξ^c$\,. In addition, we propose a simple linear algorithm for calculating $ξ^c$ of trees.
Recently in 2021, Gutman introduced the Sombor index of a graph, a novel degree-based topological index. It has been shown that the Sombor index efficiently models the thermodynamic properties of chemical compounds. Assume $\mathbb{B}_n^k$ (resp. $\mathbb{V}_n^k$) comprises all graphs with order $n$ having number of bridges (resp. vertex-connectivity) $k$. Horoldagva & Xu (2021) characterized graphs achieving the maximum Sombor index of graphs in $\mathbb{B}_n^k$. This paper characterizes graphs achieving the minimum Sombor index in $\mathbb{B}_n^k$. Certain auxiliary operation on graphs in $\…
The first geometric-arithmetic (GA) index and atom-bond connectivity (ABC) index are molecular structure descriptors which play a significant role in quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) studies. Das and Trinajstić [\textit{Chem. Phys. Lett.} \textbf{497} (2010) 149-151] showed that $GA$ index is greater than $ABC$ index for all those graphs (except $K_{1,4}$ and $T^{*}$, see Figure 1) in which the difference between maximum and minimum degree is less than or equal to 3. In this note, it is proved that $GA$ index is greater…
The general sum-connectivity index of a graph $G$ is defined as $χ_α(G)= \sum_{uv\in E(G)} (d_u + d_{v})^α$ where $d_{u}$ is degree of the vertex $u\in V(G)$, $α$ is a real number different from $0$ and $uv$ is the edge connecting the vertices $u,v$. In this note, the problem of characterizing the graphs having extremum $χ_α$ values from a certain collection of polyomino chain graphs is solved for $α<0$. The obtained results together with already known results (concerning extremum values of polyomino chain graphs) give the complete solution of the aforementioned problem.
The modified eccentric connectivity index of a graph is defined as the sum of the products of eccentricity with the total degree of neighboring vertices, over all vertices of the graph. This is a generalization of eccentric connectivity index. In this paper, we derive some upper and lower bounds for the modified eccentric connectivity index in terms of some graph parameters such as number of vertices, number of edges, radius, minimum degree, maximum degree, total eccentricity, the first and second Zagreb indices, Weiner index etc.
Let $q(G)$ denote the $Q$-index of a graph $G$, which is the largest signless Laplacian eigenvalue of $G$. We prove best possible upper bounds of $q(G)$ and best possible lower bounds of $q(\overline{G})$ for a connected graph $G$ to be $k$-connected and maximally connected, respectively. Similar upper bounds of $q(G)$ and lower bounds of $q(\overline{G})$ to assure $G$ to be super-connected are also obtained. Upper bounds of $q(G)$ and lower bounds of $q(\overline{G})$ to assure a connected triangle-free graph $G$ to be $k$-connected, maximally connected and super-connected are also respectiv…
The eccentric connectivity index of a graph $G$, denoted by $ξ^{c}(G)$, defined as $ξ^{c}(G)$ = $\sum_{v \in V(G)}ε(v) \cdot d(v)$, where $ε(v)$ and $d(v)$ denotes the eccentricity and degree of a vertex $v$ in a graph $G$, respectively. The volcano graph $V_{n,d}$ is a graph obtained from a path $P_{d+1}$ and a set $S$ of $n-d-1$ vertices, by joining each vertex in $S$ to a central vertex or vertices of $P_{d+1}$. In (A lower bound on the eccentric connectivity index of a graph, Discrete Applied Math., 160, 248 to 258, (2012)), Morgan et al. proved that $ξ^{c}(G) \geq ξ^{c}(V_{n,d})$ for any …
The {\em atom-bond connectivity (ABC) index} is a degree-based molecular descriptor, that found chemical applications. It is well known that among all connected graphs, the graphs with minimal ABC index are trees. A complete characterization of trees with minimal $ABC$ index is still an open problem. In this paper, we present new structural properties of trees with minimal ABC index. Our main results reveal that trees with minimal ABC index do not contain so-called {\em $B_k$-branches}, with $k \geq 5$, and that they do not have more than four $B_4$-branches.
The eccentric connectivity index of a graph $G$ is $ξ^c(G) = \sum_{v \in V(G)}\varepsilon(v)°(v)$, and the eccentric distance sum is $ξ^d(G) = \sum_{v \in V(G)}\varepsilon(v)D(v)$, where $\varepsilon(v)$ is the eccentricity of $v$, and $D(v)$ the sum of distances between $v$ and the other vertices. A lower and an upper bound on $ξ^d(G) - ξ^c(G)$ is given for an arbitrary graph $G$. Regular graphs with diameter at most $2$ and joins of cocktail-party graphs with complete graphs form the graphs that attain the two equalities, respectively. Sharp lower and upper bounds on $ξ^d(T) - ξ^c(T)$ are gi…
A conjecture of AutoGraphiX on the relation between the Randić index $R$ and the algebraic connectivity $a$ of a connected graph $G$ is: $$\frac R a\leq (\frac{n-3+2\sqrt{2}}{2})/(2(1- \cos {\fracπ{n}})) $$ with equality if and only if $G$ is $P_n$, which was proposed by Aouchiche and Hansen [M. Aouchiche and P. Hansen, A survey of automated conjectures in spectral graph theory, {\it Linear Algebra Appl.} {\bf 432}(2010), 2293--2322]. We prove that the conjecture holds for all trees and all connected graphs with edge connectivity $κ'(G)\geq 2$, and if $κ'(G)=1$, the conjecture holds for suffic…
For a graph $G$, $k(G)$ denotes its connectivity. A graph is super connected if every minimum vertex-cut isolates a vertex. Also $k_{1}$-connectivity of a connected graph is the minimum number of vertices whose deletion gives a disconnected graph without isolated vertices. This paper provides bounds for the super connectivity and $k_{1}$-connectivity of the lexicographic product of two graphs.
We derive a new upper bound on the algebraic connectivity of a regular graph using the Higman-Sims technique. Together with a new result on the connectivity of the neighbourhood graph of strongly regular graphs, our result gives a characterization of a class of strongly regular graphs that maximize the algebraic connectivity amongst regular graphs.
In this paper we find the vertex connectivity of Deza graphs with parameters of the complements to Seidel graphs. In particular, we present an infinite family of strictly Deza graphs whose vertex connectivity is equal to $k-1$, where $k$ is the valency.
We relate matroid connectivity to Tutte-connectivity in an infinite graph. Moreover, we show that the two cycle matroids, the finite-cycle matroid and the cycle matroid, in which also infinite cycles are taken into account, have the same connectivity function. As an application we re-prove that, also for infinite graphs, Tutte-connectivity is invariant under taking dual graphs.
Continuing the recent work of L. Zhong and K. Xu [MATCH Commun. Math. Comput. Chem.71(2014) 627-642], we determine inequalities among several vertex-degree-based topological indices; first geometric-arithmetic index(GA), augmented Zagreb index (AZI), Randi$\acute{c}$ index (R), atom-bond connectivity index (ABC), sum-connectivity index (X)and harmonic index (H).
The atom-bond connectivity (ABC) index is a degree-based molecular structure descriptor that can be used for modelling thermodynamic properties of organic chemical compounds. Motivated by its applicable potential, a series of investigations have been carried out in the past several years. In this note we first consider graphs with given edge-connectivity that attain the maximum ABC index. In particular, we give an affirmative answer to the conjecture about the structure of graphs with edge-connectivity equal to one that maximize the ABC index, which was recently raised by Zhang, Yang, Wang and…
For a (molecular) graph, the first multiplicative Zagreb index $\prod_1(G) $ is the product of the square of every vertex degree, and the second multiplicative Zagreb index $\prod_2(G) $ is the product of the products of degrees of pairs of adjacent vertices. In this paper, we explore graphs in terms of (edge) connectivity. The maximum and minimum values of $\prod_1(G) $ and $\prod_2(G) $ of graphs with connectivity at most $k$ are provided. In addition, the corresponding extremal graphs are characterized, and our results extend and enrich some known conclusions.
The {\em atom-bond connectivity (ABC) index} is a degree-based graph topological index that found chemical applications. The problem of complete characterization of trees with minimal $ABC$ index is still an open problem. In~\cite{d-sptmabci-2014}, it was shown that trees with minimal ABC index do not contain so-called {\em $B_k$-branches}, with $k \geq 5$, and that they do not have more than four $B_4$-branches. Our main results here reveal that the number of $B_1$ and $B_2$-branches are also bounded from above by small fixed constants. Namely, we show that trees with minimal ABC index do not…
Eccentric Connectivity Index of Cartesian product and Strong product of Strongly Connected Digraphs
NSS College Ottappalam（N.S.S. 奥特帕拉姆学院）；Department of Mathematics, Government College Chittur（奇图尔政府学院数学系）；University of Calicut（卡利卡特大学）；Department of Collegiate Education, Government of Kerala（喀拉拉邦 collegiate 教育部）
This paper investigates some relationship between the algebraic connectivity and the clique number of graphs. We characterize all extremal graphs which have the maximum and minimum the algebraic connectivity among all graphs of order $n$ with the clique number $r$, respectively. In turn, an upper and lower bounds for the clique number of a graph in terms of the algebraic connectivity are obtained. Moreover, a spectral version of the Erdős-Stone theorem in terms of the algebraic connectivity of graphs is presented.
Among all simple 2-connected graphs, and among all $θ$-graphs, the graphs with the minimum algebraic connectivity are completely determined, respectively.
The geometric--arithmetic indices are widely considered in the chemical graph theory in the last decade. The reason of introducing new indices is to gain prediction of target properties of considered molecules that is better than the prediction obtained by already known indices. In the case of the fifth geometric--arithmetic index hitherto no ability of prediction of some molecule property has been considered. In this paper we investigate correlations between the fifth geometric--arithmetic index and some other degree--based topological indices on the family of octane isomers and polyaromatic …
This paper considers the edge-connectivity and restricted edge-connectivity of replacement product graphs, gives some bounds on edge-connectivity and restricted edge-connectivity of replacement product graphs and determines the exact values for some special graphs. In particular, the authors further confirm that under certain conditions, the replacement product of two Cayley graphs is also a Cayley graph, and give a necessary and sufficient condition for such Cayley graphs to have maximum restricted edge-connectivity. Based on these results, the authors construct a Cayley graph with degree $d$…
The generalized $k$-connectivity $κ_k(G)$ of a graph $G$ was introduced by Chartrand et al. in 1984, which is a nice generalization of the classical connectivity. Recently, as a natural counterpart, Li et al. proposed the concept of generalized edge-connectivity for a graph. In this paper, we determine the computational complexity of the generalized connectivity and generalized edge-connectivity of a graph. Two conjectures are also proved to be true.
The power graph of a group $G$ is the graph whose vertex set is $G$ and two distinct vertices are adjacent if one is a power of the other. In this paper, the minimum degree of power graphs of certain classes of cyclic groups, abelian $p$-groups, dihedral groups and dicyclic groups are obtained. It is ascertained that the edge-connectivity and minimum degree of power graphs are equal, and consequently the minimum disconnecting sets of power graphs of the aforementioned groups are determined. Then the equality of connectivity and minimum degree of power graphs of finite groups is investigated an…
In this paper we consider the following problem: Over the class of all simple connected graphs of order $n$ with $k$ pendant vertices ($n,k$ being fixed), which graph maximizes (respectively, minimizes) the algebraic connectivity? We also discuss the algebraic connectivity of unicyclic graphs.
The generalized $k$-connectivity $κ_k(G)$ of a graph $G$, introduced by Hager in 1985, is a natural generalization of the concept of connectivity $κ(G)$, which is just for $k=2$. Total graph is generalized line graph and a large graph which obtained by incidence relation between vertices and edges of original graph. T. Hamada and T. Nonaka et al., in \cite{Hamada} determined the connectivity of the total graph $T(G)$ for a graph $G$. In this paper we determine the generalized $k$-(edge)-connectivity of total graph $T(G)$ for $k=3$.
The concept of pendant-tree connectivity, introduced by Hager in 1985, is a generalization of classical vertex-connectivity. In this paper, we study pendant-tree connectivity of line graphs.
We obtain the maximum sum-connectivity indices of graphs in the set of trees and in the set of unicyclic graphs respectively with given number of vertices and maximum degree, and determine the corresponding extremal graphs. Additionally, we deduce the n-vertex unicyclic graphs with the first two maximum sum-connectivity indices for $n\ge 4$.
The {\em atom-bond connectivity (ABC) index} is one of the recently most investigated degree-based molecular structure descriptors, that have applications in chemistry. For a graph $G$, the ABC index is defined as $\sum_{uv\in E(G)}\sqrt{\frac{(d(u) +d(v)-2)}{d(u)d(v)}}$, where $d(u)$ is the degree of vertex $u$ in $G$ and $E(G)$ is the set of edges of $G$. Despite many attempts in the last few years, it is still an open problem to characterize trees with minimal $ABC$ index. In this paper, we present an efficient approach of computing trees with minimal ABC index, by considering the degree se…
This paper deals with the vertex connectivity of enhanced power graph of finite group. We classify all abelian groups G such that vertex connectivity of enhanced power graph of G is 1. We derive an upper bound of vertex connectivity for the enhanced power graph of any general abelian group G. Also we completely characterize all abelian group G, such that the proper enhanced power graph is connected. Moreover, we study some special class of non-abelian group G such that the proper enhanced power graph is connected and we find their vertex connectivity.
In this paper we determine the unique graph with minimum distance spectral radius among all connected graphs of fixed order and given edge connectivity.
The generalized $k$-edge-connectivity $λ_k(G)$ of a graph $G$ is a generalization of the concept of edge-connectivity. The lexicographic product of two graphs $G$ and $H$, denoted by $G\circ H$, is an important graph product. In this paper, we mainly study the generalized 3-edge-connectivity of $G \circ H$, and get upper and lower bounds of $λ_3(G \circ H)$. Moreover, all bounds are sharp.
The complete set of minimal obstructions for embedding graphs into the torus is still not determined. In this paper, we present all obstructions for the torus of connectivity 2. Furthermore, we describe the building blocks of obstructions of connectivity 2 for any orientable surface.
The generalized $k$-connectivity of a graph $G$, denoted by $κ_k(G)$, is the minimum number of internally edge disjoint $S$-trees for any $S\subseteq V(G)$ and $|S|=k$. The generalized $k$-connectivity is a natural extension of the classical connectivity and plays a key role in applications related to the modern interconnection networks. The burnt pancake graph $BP_n$ and the godan graph $EA_n$ are two kinds of Cayley graphs which posses many desirable properties. In this paper, we investigate the generalized 3-connectivity of $BP_n$ and $EA_n$. We show that $κ_3(BP_n)=n-1$ and $κ_3(EA_n)=n-1$…
We investigate how the algebraic connectivity of a graph changes by relocating a connected branch from one vertex to another vertex, and then minimize the algebraic connectivity among all connected graphs of order $n$ with fixed domination number $γ\le \frac{n+2}{3}$, and finally present a lower bound for the algebraic connectivity in terms of the domination number.
Recently Ayyaswamy [1] have introduced a novel concept of the signless Laplacian Estrada index (after here $SLEE$) associated with a graph $G$. After works, we have identified the unique graph with maximum $SLEE$ with a given parameter such as: number of cut vertices, (vertex) connectivity and edge connectivity. In this paper we continue out characterization for two further parameters; diameter and number of cut vertices.
Motivated by work of Erdős, Ota determined the maximum size $g(n,k)$ of a $k$-connected nonhamiltonian graph of order $n$ in 1995. But for some pairs $n,k,$ the maximum size is not attained by a graph of connectivity $k.$ For example, $g(15,3)=77$ is attained by a unique graph of connectivity $7,$ not $3.$ In this paper we obtain more precise information by determining the maximum size of a nonhamiltonian graph of order $n$ and connectivity $k,$ and determining the extremal graphs. Consequently we solve the corresponding problem for nontraceable graphs.
The generalized $k$-connectivity $κ_{k}(G)$ of a graph $G$, which was introduced by Chartrand et al.(1984) is a generalization of the concept of vertex connectivity. Let $G$ and $H$ be nontrivial connected graphs. Recently, Li et al. gave a lower bound for the generalized $3$-connectivity of the Cartesian product graph $G \square H$ and proposed a conjecture for the case that $H$ is $3$-connected. In this paper, we give two different forms of lower bounds for the generalized $3$-connectivity of Cartesian product graphs. The first lower bound is stronger than theirs, and the second confirms the…
We study the algebraic connectivity (or second Laplacian eigenvalue) of token graphs, also called symmetric powers of graphs. The $k$-token graph $F_k(G)$ of a graph $G$ is the graph whose vertices are the $k$-subsets of vertices from $G$, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in $G$. Recently, it was conjectured that the algebraic connectivity of $F_k(G)$ equals the algebraic connectivity of $G$. In this paper, we prove the conjecture for new infinite families of graphs, such as trees and graphs with maximum degree large enough.
Dirac showed that in a $(k-1)$-connected graph there is a path through each $k$ vertices. The path $k$-connectivity $π_k(G)$ of a graph $G$, which is a generalization of Dirac's notion, was introduced by Hager in 1986. In this paper, we study path connectivity of line graphs.
A conjecture of Luo, Tian and Wu (2022) says that for every positive integer $k$ and every finite tree $T$ with bipartition $X$ and $Y$ (denote $t = \max\{|X|,|Y |\})$, every $k$-connected bipartite graph $G$ with $δ(G) \geq k + t$ contains a subtree $T' \cong T$ such that $κ(G-V (T')) \geq k$. In this paper, we confirm this conjecture for caterpillars when $k=3$ and spiders when $k\leq 3$.
The generalized $k$-connectivity $κ_k(G)$ of a graph $G$, introduced by Hager in 1985, is a nice generalization of the classical connectivity. Recently, as a natural counterpart, we proposed the concept of generalized $k$-edge-connectivity $λ_k(G)$. In this paper, graphs of order $n$ such that $κ_k(G)=n-\frac{k}{2}-1$ and $λ_k(G)=n-\frac{k}{2}-1$ for even $k$ are characterized.
We introduce the concept of matching connectivity as a notion of connectivity in graph admitting perfect matchings which heavily relies on the structural properties of those matchings. We generalise a result of Robertson, Seymour and Thomas for bipartite graphs with perfect matchings (see [Neil Roberts, Paul D Seymour, and Robin Thomas. Permanents, pfaffian orientations, and even directed curcuits. Annals of Mathematics, 150(2):929-975, 1999]) in order to obtain a concept of alternating paths that turns out to be sufficient for the description of our connectivity parameter. We introduce some b…
We give necessary and sufficient conditions for lobe-transitivity of locally finite and locally countable graphs whose connectivity equals 1. We show further that, given any biconnected graph $Λ$ and a "code" assigned to each orbit of Aut($Λ$), there exists a unique lobe-transitive graph $Γ$ of connectivity 1 whose lobes are copies of $Λ$ and is consistent with the given code at every vertex of $Γ$. These results lead to necessary and sufficient conditions for a graph of connectivity $1$ to be edge-transitive and to be arc-transitive. Countable graphs of connectivity 1 the action of whose auto…
