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 given for arbitrary trees. Sharp lower and upper bounds on $ξ^d(G)+ξ^c(G)$ for arbitrary graphs $G$ are also given, and a sharp lower bound on $ξ^d(G)$ for graphs $G$ with a given radius is proved.
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.
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 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 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.
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 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.
The eccentric-connectivity index of a graph G is the sum of the products of the eccentricity and the degree of each vertex in G. In this paper, we define four new invariants related to the eccentric-connectivity index and obtain upper bounds for total transformation graphs which are some generalizations of total graph.
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）
In a connected graph G, the distance between two vertices of G is the length of a shortest path between these vertices. The eccentricity of a vertex u in G is the largest distance between u and any other vertex of G. The total-eccentricity index τ(G) is the sum of eccentricities of all vertices of G. In this paper, we find extremal trees, unicyclic and bicyclic graphs with respect to total-eccentricity index. Moreover, we find extremal conjugated trees with respect to total-eccentricity index.
The connective eccentricity index $ξ^{ce}=\sum^{}_{u\in V}\frac{d(u)}{\varepsilon(u)}$, where $\varepsilon(u)$ and $d(u)$ denote the eccentricity and the degree of the vertex $u$, respectively. In this paper, we first determine the extremal trees which minimize and maximize the connective eccentricity index among all trees with a given degree sequence, and then determine the extremal trees which minimize and maximize the connective eccentricity index among all trees with a given number of branching vertices.
The eccentricity of a vertex $u$ in a connected graph $G$ is the distance between $u$ and a vertex farthest from it; the eccentric sequence of $G$ is the nondecreasing sequence of the eccentricities of $G$. In this paper, we determine the unique tree that minimises the Wiener index, i.e. the sum of distances between all unordered vertex pairs, among all trees with a given eccentric sequence. We show that the same tree maximises the number of subtrees among all trees with a given eccentric sequence, thus providing another example of negative correlation between the number of subtrees and the Wi…
Topological indices are molecular descriptors that describe the properties of chemical compounds. These topological indices correlate specific physico-chemical properties like boiling point, enthalpy of vaporization, strain energy, and stability of chemical compounds. This article introduces a new graph operation based on central graph called central vertex-edge join and provides its results related to graph invariants like eccentric-connectivity index, connective eccentricity index, total-eccentricity index, average eccentricity index, Zagreb eccentricity indices, eccentric geometric-arithmet…
The total eccentricity index of a connected graph is defined as sum of the eccentricities of all its vertices. We denote the set of all connected graphs on $n$ vertices with $k$ pendant vertices by $\mathfrak{H}_{n,k}$ and denote the set of all connected graphs on $n$ vertices with $s$ cut vertices by $\mathfrak{C_{n,s}}$. In this paper, we give the sharp lower and upper bounds on the total eccentricity index over $\mathfrak{H}_{n,k}$ and the sharp lower bound for the same over $\mathfrak{C_{n,s}}$. We also provide the sharp upper bounds on the total eccentricity index over $\mathfrak{C_{n,s}}…
The relation between the Wiener index $W(G)$ and the eccentricity $\varepsilon(G)$ of a graph $G$ is studied. Lower and upper bounds on $W(G)$ in terms of $\varepsilon(G)$ are proved and extremal graphs characterized. A Nordhaus-Gaddum type result on $W(G)$ involving $\varepsilon(G)$ is given. A sharp upper bound on the Wiener index of a tree in terms of its eccentricity is proved. It is shown that in the class of trees of the same order, the difference $W(T) - \varepsilon(T)$ is minimized on caterpillars. An exact formula for $W(T) - \varepsilon(T)$ in terms of the radius of a tree $T$ is obt…
The first and the second Zagreb eccentricity index of a graph $G$ are defined as $E_1(G)=\sum_{v\in V(G)}\varepsilon_{G}(v)^{2}$ and $E_2(G)=\sum_{uv\in E(G)}\varepsilon_{G}(u)\varepsilon_{G}(v)$, respectively, where $\varepsilon_G(v)$ is the eccentricity of a vertex $v$. In this paper the invariants $E_1$, $E_2$, and the Wiener index are compared on graphs with diameter $2$, on trees, on a newly introduced class of universally diametrical graphs, and on Cartesian product graphs. In particular, if the diameter of a tree $T$ is not too big, then $W(T) \ge E_2(T)$ holds, and if the diameter of $…
For a connected graph $G$, the Wiener index, denoted by $W(G)$, is the sum of the distance of all pairs of distinct vertices and the eccentricity, denoted by $\varepsilon(G)$, is the sum of the eccentricity of individual vertices. In \cite{Kc}, the authors posed a conjecture which states that given a graph $G$ with at least three vertices, the difference between $W(G)$ and $\varepsilon(G)$ decreases when an edge is contracted and proved that the conjecture is true when $e$ is a bridge. In this manuscript, we confirm that the conjecture is true for any connected graph $G$ with at least three ve…
The connective eccentricity index (CEI) of a graph $G$ is defined as $ξ^{ce}(G)=\sum_{v \in V(G)}\frac{d_G(v)}{\varepsilon_G(v)}$, where $d_G(v)$ is the degree of $v$ and $\varepsilon_G(v)$ is the eccentricity of $v$. In this paper, we characterize the unique graphs with maximum CEI from three classes of graphs: the $n$-vertex graphs with fixed connectivity and diameter, the $n$-vertex graphs with fixed connectivity and independence number, and the $n$-vertex graphs with fixed connectivity and minimum degree.
The eccentric sequence of a connected graph $G$ is the nondecreasing sequence of the eccentricities of its vertices. The Wiener index of $G$ is the sum of the distances between all unordered pairs of vertices of $G$. The unique trees that minimise the Wiener index among all trees with a given eccentric sequence were recently determined by the present authors. In this paper we show that these results hold not only for the Wiener index, but for a large class of distance-based topological indices which we term Wiener-type indices. Particular cases of this class include the hyper-Wiener index, the…
The connective eccentric index of a graph is a topological index involving degrees and eccentricities of vertices of the graph. In this paper, we have studied the connective eccentric index for double graph and double cover. Also we give the connective eccentric index for some graph operations such as joins, symmetric difference, disjunction and splice of graphs.
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.
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.
We conjecture a new lower bound on the algebraic connectivity of a graph that involves the number of vertices of high eccentricity in a graph. We prove that this lower bound implies a strengthening of the Laplacian Spread Conjecture. We discuss further conjectures, also strengthening the Laplacian Spread Conjecture, that include a conjecture for simple graphs and a conjecture for weighted graphs.
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 vertex $k$-partiteness $v_k(G)$ of graph $G$ is defined as the fewest number of vertices whose deletion from $G$ yields a $k$-partite graph. In this paper, we introduce two concepts: monotonic decreasing topological index and monotonic increasing topological index, and characterize the extremal graphs having the minimum Wiener index, the maximum Harry index, the maximum reciprocal degree distance, the minimum eccentricity distance sum, the minimum adjacent eccentric distance sum index, the maximum connective eccentricity index, the maximum Zagreb indices among graphs with a fixed number $n…
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
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.
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.
Let $G$ be a connected edge-weighted graph of order $n$ and size $m$. Let $w:E(G)\rightarrow \mathbb{R}^{\geq 0}$ be the weighting function. We assume that $w$ is normalised, that is, $\sum_{e\in E(G)} w(e)=m$. The weighted distance $d_w(u,v)$ between any two vertices $u$ and $v$ is the least weight between them and the eccentricity $e_w(v)$ of a vertex $v$ is the weighted distance from $v$ to a vertex farthest from it in $G$. The mean(average) eccentricity of $G$, $avec(G,w)$, is the (weighted) mean of all eccentricities in $G$. We obtain upper and lower bounds on $avec(G,w)$ in terms of $n$,…
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…
Let $I(G)$ be a topological index of a graph. If $I(G+e)<I(G)$ (or $I(G+e)>I(G)$, respectively) for each edge $e\not\in G$, then $I(G)$ is monotonically decreasing (or increasing, respectively) with the addition of edges. In this article, we present lower or upper bounds for some monotonic topological indices, including the Wiener index, the hyper-Wiener index, the Harary index, the connective eccentricity index, the eccentricity distance sum of bipartite graphs in terms of the number of cut edges, and characterize the corresponding extremal graphs, respectively.
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…
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 {\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.
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.
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.
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…
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…
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 …
For a graph $G$, the first multiplicative Zagreb index $\prod_1(G) $ is the product of squares of vertex degrees, and the second multiplicative Zagreb index $\prod_2(G) $ is the product of products of degrees of pairs of adjacent vertices. In this paper, we explore graphs with extremal $Π_{1}(G)$ and $Π_{2}(G)$ in terms of (edge) connectivity and pendant vertices. The corresponding extremal graphs are characterized with given connectivity at most $k$ and $p$ pendant vertices. In addition, the maximum and minimum values of $\prod_1(G) $ and $\prod_2(G) $ are provided. Our results extend and enr…
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…
The Sombor index is one of the geometry-based descriptors, which was defined as $$SO(G)=\sum_{uv\in E(G)}\sqrt{d^{2}(u)+d^{2}(v)},$$ where $d(u)$ (resp. $d(v)$) denotes the degree of vertex $u$ (resp. $v$) in $G$. In this note, we determine the maximum and minimum graphs with respect to the Sombor index among the set of graphs with vertex connectivity (resp. edge connectivity) at most $k$, which solves an open problem on the Sombor index proposed by Hayat and Rehman [On Sombor index of graphs with a given number of cut-vertices, MATCH Commun. Math. Comput. Chem. 89 (2023) 437--450]. For some o…
We present the bounds in terms of other important graph parameters for general Sombor index which generalises both the forgotten index and the Sombor index. We also explore the Nordhaus-Gaddum-type result for the general Sombor index. We present further the relations between general Sombor index and other generalised indices: general Randić index and general sum-connectivity index.
The Estrada index of a graph $G$ is defined as $EE(G)=\sum_{i=1}^ne^{λ_i}$, where $λ_1,$ $ λ_2,\ldots, λ_n$ are the eigenvalues of the adjacency matrix of $G$. In this paper, we characterize the unique bipartite graph with maximum Estrada index among bipartite graphs with given matching number and given vertex-connectivity, edge-connectivity, respectively.
The Randic (connectivity) index is one of the most successful molecular descriptors in structure-property and structure-activity relationships studies. J. Gao found the sharp upper bound for the Randic index of apex trees. In this paper, we proved that k-apex trees are not regular graphs for k\ge2 and proposed a sharp upper bound for the Randic index of k-apex trees for k>1.
The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. We provide formulae for the minimum Wiener index of simple triangulations and quadrangulations with connectivity at least $c$, and provide the extremal structures, which attain those values. Our main tool is setting upper bounds for the maximum degree in highly connected triangulations and quadrangulations.
The first Zagreb index $M_{1}$ of a graph is defined as the sum of the square of every vertex degree, and the second Zagreb index $M_{2}$ of a graph is defined as the sum of the product of vertex degrees of each pair of adjacent vertices. In this paper, we study the Zagreb indices of bipartite graphs of order $n$ with $κ(G)=k$ (resp. $κ'(G)=s$) and sharp upper bounds are obtained for $M_1(G)$ and $M_2(G)$ for $G\in \mathcal{V}^k_n$ (resp. $\mathcal{E}^s_n$), where $\mathcal{V}^k_n$ is the set of bipartite graphs of order $n$ with $κ(G)=k$, and $\mathcal{E}^s_n$ is the set of bipartite graphs o…
