The cyclic edge-connectivity of a graph $G$ is the least $k$ such that there exists a set of $k$ edges whose removal disconnects $G$ into components where every component contains a cycle. We show that for graphs of minimum degree at least 3 and girth $g$ at least 4, the cyclic edge-connectivity is bounded above by $(Δ-2)g$ where $Δ$ is the maximum degree. We then prove that if the second eigenvalue of the adjacency matrix of a $d$-regular graph of girth $g\geq4$ is sufficiently small, then the cyclic edge-connectivity is $(d-2)g$, providing a spectral condition for when this upper bound on cyclic edge-connectivity is tight.
Let $G$ be a cyclically $5$-connected cubic graph with a $5$-edge-cut separating $G$ into two cyclic components $G_1$ and $G_2$. We prove that each component $G_i$ can be completed to a cyclically $5$-connected cubic graph by adding three vertices, unless $G_i$ is a cycle of length five. Our work extends similar results by Andersen et al. for cyclic connectivity $4$ from 1988.
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 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…
A graph is said to be cyclic $k$-edge-connected, if at least $k$ edges must be removed to disconnect it into two components, each containing a cycle. Such a set of $k$ edges is called a cyclic-$k$-edge cutset and it is called a trivial cyclic-$k$-edge cutset if at least one of the resulting two components induces a single $k$-cycle. It is known that fullerenes, that is, 3-connected cubic planar graphs all of whose faces are pentagons and hexagons, are cyclic 5-edge-connected. In this article it is shown that a fullerene $F$ containing a nontrivial cyclic-5-edge cutset admits two antipodal pent…
A path in a vertex-colored graph is called a \emph{vertex-monochromatic path} if its internal vertices have the same color. A vertex-coloring of a graph is a \emph{monochromatic vertex-connection coloring} (\emph{MVC-coloring} for short), if there is a vertex-monochromatic path joining any two vertices in the graph. For a connected graph $G$, the \emph{monochromatic vertex-connection number}, denoted by $mvc(G)$, is defined to be the maximum number of colors used in an \emph{MVC-coloring} of $G$. These concepts of vertex-version are natural generalizations of the colorful monochromatic connect…
We survey various aspects of infinite extremal graph theory and prove several new results. The lead role play the parameters connectivity and degree. This includes the end degree. Many open problems are suggested.
The concept of maximum local connectivity $\bar κ$ of a graph was introduced by Bollobás. One of the problems about it is to determine the largest number of edges $f(n;\barκ\leq \ell)$ for graphs of order $n$ that have local connectivity at most $\ell$. We consider a generalization of the above concept and problem. For $S\subseteq V(G)$ and $|S|\geq 2$, the \emph{generalized local connectivity} $κ(S)$ is the maximum number of internally disjoint trees connecting $S$ in $G$. The parameter $\barκ_k(G)=max\{κ(S)|S\subseteq V(G),|S|=k\}$ is called the \emph{maximum generalized local connectivity} …
Let $n=p_1^{n_1}p_2^{n_2}\ldots p_r^{n_r}$, where $r,n_1,\ldots, n_r$ are positive integers and $p_1,p_2,\ldots,p_r$ are distinct prime numbers with $p_1<p_2<\cdots <p_r$. For the cyclic group $C_n$ of order $n$, let $\mathcal{P}(C_n)$ be the power graph of $C_n$ and $κ(\mathcal{P}(C_n))$ be the vertex connectivity of $\mathcal{P}(C_n)$. It is known that $κ(\mathcal{P}(C_n))=p_1^{n_1} -1$ if $r=1$. For $r\geq 2$, we determine the exact value of $κ(\mathcal{P}(C_n))$ when $2ϕ(p_1\ldots p_{r-1})\geq p_1\ldots p_{r-1}$, and give an upper bound for $κ(\mathcal{P}(C_n))$ when $2ϕ(p_1\ldots p_{r-1})…
For a symmetric bivariable function $f(x,y)$, let the {\it connectivity function} of a connected graph $G$ be $M_f(G)=\sum_{uv\in E(G)}f(d(u),d(v))$, where $d(u)$ is the degree of vertex $u$. In this paper, we prove that for an escalating (de-escalating) function $f(x,y)$, there exists a BFS-graph with the maximum (minimum) connectivity function $M_f(G)$ among all graphs with a $c-$cyclic degree sequence $π=(d_1,d_2, \ldots, d_n)$ and $d_n=1$, and obtain the majorization theorem for connectivity function for unicyclic and bicyclic degree sequences. Moreover, some applications of graph invarian…
For a digraph $D=(V(D), A(D))$, and a set $S\subseteq V(D)$ with $r\in S$ and $|S|\geq 2$, an $(S, r)$-tree is an out-tree $T$ rooted at $r$ with $S\subseteq V(T)$. Two $(S, r)$-trees $T_1$ and $T_2$ are said to be arc-disjoint if $A(T_1)\cap A(T_2)=\emptyset$. Two arc-disjoint $(S, r)$-trees $T_1$ and $T_2$ are said to be internally disjoint if $V(T_1)\cap V(T_2)=S$. Let $κ_{S,r}(D)$ and $λ_{S,r}(D)$ be the maximum number of internally disjoint and arc-disjoint $(S, r)$-trees in $D$, respectively. The generalized $k$-vertex-strong connectivity of $D$ is defined as $$κ_k(D)= \min \{κ_{S,r}(D)\…
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 power graph $\mathcal{P}(G)$ of a given finite group $G$ is the simple undirected graph whose vertices are the elements of $G$, in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. The vertex connectivity $κ(\mathcal{P}(G))$ of $\mathcal{P}(G)$ is the minimum number of vertices which need to be removed from $G$ so that the induced subgraph of $\mathcal{P}(G)$ on the remaining vertices is disconnected or has only one vertex. For a positive integer $n$, let $C_n$ be the cyclic group of order $n$. Suppose that the prime powe…
Extremal properties and bounds for the generalized algebraic connectivity of graphs in Euclidean spaces
Universidad de Buenos Aires（布宜诺斯艾利斯大学）；CONICET-Universidad de Buenos Aires, INTECIN（阿根廷国家科学研究委员会-布宜诺斯艾利斯大学，智能通信研究所）；Universidad de San Andrés（圣安德烈斯大学）；CONICET（阿根廷国家科学研究委员会）
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 a 1983 paper, G. Ramharter asks what are the extremal arrangements for the cyclic analogues of the regular and semi-regular continuants first introduced by T.S. Motzkin and E.G. Straus in 1956. In this paper we answer this question by showing that for each set $A$ consisting of positive integers $1<a_1<a_2<\cdots <a_k$ and a $k$-term partition $P: n_1+n_2 + \cdots + n_k=n$, there exists a unique (up to reversal) cyclic word $x$ which maximizes (resp. minimizes) the regular cyclic continuant $K^{\circlearrowright}(\cdot)$ amongst all cyclic words over $A$ with Parikh vector $(n_1,n_2,\ldots,…
Boesch and Chen (SIAM J. Appl. Math., 1978) introduced the cut-version of the generalized edge-connectivity, named $k$-edge-connectivity. For any integer $k$ with $2\leq k\leq n$, the {\em $k$-edge-connectivity} of a graph $G$, denoted by $λ_k(G)$, is defined as the smallest number of edges whose removal from $G$ produces a graph with at least $k$ components. In this paper, we first compute some exact values and sharp bounds for $λ_k(G)$ in terms of $n$ and $k$. We then discuss the relationships between $λ_k(G)$ and other generalized connectivities. An algorithm in $\mathcal{O}(n^2)$ time will…
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 $n\ge 4$.
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.
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.
A $k$-cyclic graph is a connected graph of order $n$ and size $n+k-1$. In this paper, we determine the maximal signless Laplacian spectral radius and the corresponding extremal graph among all $C_{4}$-free $k$-cyclic graphs of order $n$. Furthermore, we determine the first three unicyclic, and bicyclic, $C_{4}$-free graphs whose spectral radius of the signless Laplacian is maximal. Similar results are obtained for the (combinatorial) Laplacian.
The generalization of the Oberwolfach Problem, proposed by J. Liu in 2000, asks for a uniform $2$-factorization of the complete multipartite graph $K_{m\times n}$. Here we focus our attention on $2$-factorizations regular under the cyclic group $Z_{mn}$, whose $2$-factors are disjoint union of cycles all of even length $\ell$. In particular, we present a complete solution for the extremal cases $\ell=4$ and $\ell=mn$.
The problem of determining the largest number $f(n;\barκ\leq \ell)$ of edges for graphs with $n$ vertices and maximal local connectivity at most $\ell$ was considered by Bollobás. Li et al. studied the largest number $f(n;\barκ_3\leq2)$ of edges for graphs with $n$ vertices and at most two internally disjoint Steiner trees connecting any three vertices. In this paper, we further study the largest number $f(n;\barκ_k=1)$ of edges for graphs with $n$ vertices and exactly one Steiner tree connecting any $k$ vertices with $k\geq 3$. It turns out that this is not an easy task to finish, not like th…
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.
For an edge-colored graph $G$, we call an edge-cut $M$ of $G$ monochromatic if the edges of $M$ are colored with the same color. The graph $G$ is called monochromatic disconnected if any two distinct vertices of $G$ are separated by a monochromatic edge-cut. For a connected graph $G$, the monochromatic disconnection number (or $MD$-number for short) of $G$, denoted by $md(G)$, is the maximum number of colors that are allowed in order to make $G$ monochromatic disconnected. For graphs with diameter one, they are complete graphs and so their $MD$-numbers are $1$. For graphs with diameter at leas…
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.
Extremal spectral gap of regular graphs with bounded vertex connectivity
College of Mathematics and System Science, Xinjiang University（新疆大学数学与系统科学学院）；School of Mathematics and Statistics, The University of Melbourne（墨尔本大学数学与统计学院）
The family of snarks -- connected bridgeless cubic graphs that cannot be 3-edge-coloured -- is well-known as a potential source of counterexamples to several important and long-standing conjectures in graph theory. These include the cycle double cover conjecture, Tutte's 5-flow conjecture, Fulkerson's conjecture, and several others. One way of approaching these conjectures is through the study of structural properties of snarks and construction of small examples with given properties. In this paper we deal with the problem of determining the smallest order of a nontrivial snark (that is, one w…
Let $G$ be a (multi)graph of order $n$ and let $u,v$ be vertices of $G$. The maximum number of internally disjoint $u$-$v$ paths in $G$ is denoted by $κ_G(u,v)$, and the maximum number of edge-disjoint $u$-$v$ paths in $G$ is denoted by $λ_G (u,v)$. The average connectivity of $G$ is defined by $\overlineκ(G)=\sum_{\{u,v\}\subseteq V(G)} κ_G(u,v)/\tbinom{n}{2},$ and the average edge-connectivity of $G$ is defined by $\overlineλ(G)=\sum_{\{u,v\}\subseteq V(G)} λ_G(u,v)/\tbinom{n}{2}$. A graph $G$ is called ideally connected if $κ_G(u,v)=\min\{\mathrm{deg}(u),\mathrm{deg}(v)\}$ for all pairs of …
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.
We estimate the minimum number of vertices of a cubic graph with given oddness and cyclic connectivity. We prove that a bridgeless cubic graph $G$ with oddness $ω(G)$ other than the Petersen graph has at least $5.41\cdotω(G)$ vertices, and for each integer $k$ with $2\le k\le 6$ we construct an infinite family of cubic graphs with cyclic connectivity $k$ and small oddness ratio $|V(G)|/ω(G)$. In particular, for cyclic connectivity 2, 4, 5, and 6 we improve the upper bounds on the oddness ratio of snarks to 7.5, 13, 25, and 99 from the known values 9, 15, 76, and 118, respectively. In addition,…
Connectivity and diagnosability are two important parameters for the fault tolerant of an interconnection network $G$. In 1996, Fàbrega and Fiol proposed the $g$-good-neighbor connectivity of $G$. In this paper, we show that $1\leq κ^g(G)\leq n-2g-2$ for $0\leq g\leq \left\{Δ(G),\left\lfloor \frac{n-3}{2}\right\rfloor\right\}$, and graphs with $κ^g(G)=1,2$ and trees with $κ^g(T_n)=n-t$ for $4\leq t\leq \frac{n+2}{2}$ are characterized, respectively. In the end, we get the three extremal results for the $g$-good-neighbor connectivity.
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 eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity $ecc (G)$ of a graph $G$ is the mean value of eccentricities of all vertices of $G$. The average eccentricity is deeply connected with a topological descriptor called the eccentric connectivity index, defined as a sum of products of vertex degrees and eccentricities. In this paper we analyze extremal properties of the average eccentricity, introducing two graph transformations that increase or decrease $ecc (G)$. Furthermore, we resolve four conjectures, obtained by the system AutoGraphi…
The power graph of a group is the simple graph with vertices as the group elements, in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. We study (minimal) cut-sets of the power graph of a (finite) non-cyclic (nilpotent) group which are associated with its maximal cyclic subgroups. Let $G$ be a finite non-cyclic nilpotent group whose order is divisible by at least two distinct primes. If $G$ has a Sylow subgroup which is neither cyclic nor a generalized quaternion $2$-group and all other Sylow subgroups of $G$ are cyclic, the…
Let $D=(V,A)$ be a digraphs without isolated vertices. A vertex-degree based invariant $I(D)$ related to a real function $φ$ of $D$ is defined as a summation over all arcs, $I(D) = \frac{1}{2}\sum_{uv\in A}{φ(d_u^+,d_v^-)}$, where $d_u^+$ (resp. $d_u^-$) denotes the out-degree (resp. in-degree) of a vertex $u$. In this paper, we give the extremal values and extremal digraphs of $I(D)$ over all digraphs with $n$ non-isolated vertices. Applying these results, we obtain the extremal values of some vertex-degree based topological indices of digraphs, such as the Randić index, the Zagreb index, the…
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…
Connectivity and diagnosability are two important parameters for the fault tolerant of an interconnection network $G$. In 1996, Fàbrega and Fiol proposed the $g$-extra connectivity of $G$. A subset of vertices $S$ is said to be a \emph{cutset} if $G-S$ is not connected. A cutset $S$ is called an \emph{$R_g$-cutset}, where $g$ is a non-negative integer, if every component of $G-S$ has at least $g+1$ vertices. If $G$ has at least one $R_g$-cutset, the \emph{$g$-extra connectivity} of $G$, denoted by $κ_g(G)$, is then defined as the minimum cardinality over all $R_g$-cutsets of $G$. In this paper…
The problem of determining the smallest number of edges, $h(n;\barκ\geq r)$, which guarantees that any graph with $n$ vertices and $h(n;\barκ\geq r)$ edges will contain a pair of vertices joined by $r$ internally disjoint paths was posed by Erdös and Gallai. Bollobás considered the problem of determining the largest number of edges $f(n;\barκ\leq \ell)$ for graphs with $n$ vertices and local connectivity at most $\ell$. One can see that $f(n;\barκ\leq \ell)= h(n;\barκ\geq \ell+1)-1$. These two problems had received a wide attention of many researchers in the last few decades. In the above prob…
The generalized $k$-connectivity $κ_k(G)$ of a graph $G$ was introduced by Hager before 1985. As its a natural counterpart, we introduced the concept of generalized edge-connectivity $λ_k(G)$, recently. In this paper we summarize the known results on the generalized connectivity and generalized edge-connectivity. After an introductory section, the paper is then divided into nine sections: the generalized (edge-)connectivity of some graph classes, algorithms and computational complexity, sharp bounds of $κ_k(G)$ and $λ_k(G)$, graphs with large generalized (edge-)connectivity, Nordhaus-Gaddum-ty…
