scholarly journals Graph weights arising from Mayer and Ree-Hoover theories of virial expansions

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Amel Kaouche ◽  
Pierre Leroux
Keyword(s):  
Connected Graph ◽  
Convex Polytopes ◽  
Hard Core ◽  
Ideal Gas ◽  
One Dimension ◽  
Graph Invariants ◽  
Explicit Formulas ◽  
Connected Graphs ◽  
International Audience

International audience We study graph weights (i.e., graph invariants) which arise naturally in Mayer's theory and Ree-Hoover's theory of virial expansions in the context of a non-ideal gas. We give special attention to the Second Mayer weight $w_M(c)$ and the Ree-Hoover weight $w_{RH}(c)$ of a $2$-connected graph $c$ which arise from the hard-core continuum gas in one dimension. These weights are computed using signed volumes of convex polytopes naturally associated with the graph $c$. Among our results are the values of Mayer's weight and Ree-Hoover's weight for all $2$-connected graphs $b$ of size at most $8$, and explicit formulas for certain infinite families. Nous étudions les poids de graphes (c'est-à-dire, les invariants de graphes) qui apparaissent naturellement dans la théorie de Mayer et la théorie de Ree-Hoover pour le développement du viriel dans le contexte d'un gaz imparfait. Nous donnons une attention particulière au deuxième poids $w_M(c)$ de Mayer et au poids $w_{RH}(c)$ de Ree-Hoover d'un graphe $2$-connexe $c$ dans le cas d'un gaz à noyaux durs et à positions continues en une dimension. Ces poids sont calculés à partir de volumes signés de polytopes convexes associés naturellement au graphe $c$. Parmi nos résultats sont les valeurs du poids de Mayer et du poids de Ree-Hoover pour tous les graphes $2$-connexes $b$ de taille au plus $8$, et des formules explicites pour certaines familles infinies.

scholarly journals Cyclic sieving phenomenon in non-crossing connected graphs

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Alan Guo
Keyword(s):  
Generating Function ◽  
Connected Graph ◽  
Connected Graphs ◽  
Cyclic Sieving Phenomenon ◽  
International Audience ◽  
Cyclic Sieving

International audience A non-crossing connected graph is a connected graph on vertices arranged in a circle such that its edges do not cross. The count for such graphs can be made naturally into a q-binomial generating function. We prove that this generating function exhibits the cyclic sieving phenomenon, as conjectured by S.-P. Eu. Un graphe connexe dont les sommets sont disposés sur un cercle est sans croisement si ses arêtes ne se croisent pas. Nous démontrons une conjecture de S.-P. Eu affirmant que la fonction génératrice q-binomiale dénombrant de tels graphes exhibe le phénomène du crible cyclique.

scholarly journals Zagreb Eccentricity Indices of the Generalized Hierarchical Product Graphs and Their Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhaoyang Luo ◽  
Jianliang Wu
Keyword(s):  
Connected Graph ◽  
Explicit Formulas ◽  
Connected Graphs ◽  
Product Graphs ◽  
Corona Product

LetGbe a connected graph. The first and second Zagreb eccentricity indices ofGare defined asM1*(G)=∑v∈V(G)‍εG2(v)andM2*(G)=∑uv∈E(G)‍εG(u)εG(v), whereεG(v)is the eccentricity of the vertexvinGandεG2(v)=(εG(v))2. Suppose thatG(U)⊓H(∅≠U⊆V(G))is the generalized hierarchical product of two connected graphsGandH. In this paper, the Zagreb eccentricity indicesM1*andM2*ofG(U)⊓Hare computed. Moreover, we present explicit formulas for theM1*andM2*ofS-sum graph, Cartesian, cluster, and corona product graphs by means of some invariants of the factors.

scholarly journals Approximation and inapproximability results on balanced connected partitions of graphs

2007 ◽  
Vol Vol. 9 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Frédéric Chataigner ◽  
Liliane R. B. Salgado ◽  
Yoshiko Wakabayashi
Keyword(s):  
Approximation Algorithms ◽  
Positive Integer ◽  
Approximation Algorithm ◽  
Weight Function ◽  
Connected Graph ◽  
Strong Sense ◽  
Connected Graphs ◽  
International Audience ◽  
Arbitrary Graphs ◽  
Balanced Connected Partition

Graphs and Algorithms International audience Let G=(V,E) be a connected graph with a weight function w: V \to \mathbbZ₊, and let q ≥q 2 be a positive integer. For X⊆ V, let w(X) denote the sum of the weights of the vertices in X. We consider the following problem on G: find a q-partition P=(V₁,V₂, \ldots, V_q) of V such that G[V_i] is connected (1≤q i≤q q) and P maximizes \rm min\w(V_i): 1≤q i≤q q\. This problem is called \textitMax Balanced Connected q-Partition and is denoted by BCP_q. We show that for q≥q 2 the problem BCP_q is NP-hard in the strong sense, even on q-connected graphs, and therefore does not admit a FPTAS, unless \rm P=\rm NP. We also show another inapproximability result for BCP₂ on arbitrary graphs. On q-connected graphs, for q=2 the best result is a \frac43-approximation algorithm obtained by Chleb\'ıková; for q=3 and q=4 we present 2-approximation algorithms. When q is not fixed (it is part of the instance), the corresponding problem is called \textitMax Balanced Connected Partition, and denoted as BCP. We show that BCP does not admit an approximation algorithm with ratio smaller than 6/5, unless \rm P=\rm NP.

scholarly journals A Partition of Connected Graphs

10.37236/1968 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Gus Wiseman
Keyword(s):  
Connected Graph ◽  
Simple Structure ◽  
Chromatic Polynomial ◽  
Graph Invariants ◽  
Connected Graphs ◽  
Vertex Set ◽  
Broken Circuit

We define an algorithm $k$ which takes a connected graph $G$ on a totally ordered vertex set and returns an increasing tree $R$ (which is not necessarily a subtree of $G$). We characterize the set of graphs $G$ such that $k(G)=R$. Because this set has a simple structure (it is isomorphic to a product of non-empty power sets), it is easy to evaluate certain graph invariants in terms of increasing trees. In particular, we prove that, up to sign, the coefficient of $x^q$ in the chromatic polynomial $\chi_G(x)$ is the number of increasing forests with $q$ components that satisfy a condition that we call $G$-connectedness. We also find a bijection between increasing $G$-connected trees and broken circuit free subtrees of $G$.

scholarly journals On minimum algebraic connectivity of graphs whose complements are bicyclic

2019 ◽  
Vol 17 (1) ◽  
pp. 1490-1502 ◽  
Author(s):  
Jia-Bao Liu ◽  
Muhammad Javaid ◽  
Mohsin Raza ◽  
Naeem Saleem
Keyword(s):  
Connected Graph ◽  
Diffusion Processes ◽  
Laplacian Matrix ◽  
Group Differences ◽  
Algebraic Connectivity ◽  
Connected Graphs ◽  
Bicyclic Graph ◽  
Smallest Eigenvalue ◽  
Unique Graph ◽  
The Stability

Abstract The second smallest eigenvalue of the Laplacian matrix of a graph (network) is called its algebraic connectivity which is used to diagnose Alzheimer’s disease, distinguish the group differences, measure the robustness, construct multiplex model, synchronize the stability, analyze the diffusion processes and find the connectivity of the graphs (networks). A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. In this paper, firstly the unique graph with a minimum algebraic connectivity is characterized in the class of connected graphs whose complements are bicyclic with exactly three cycles. Then, we find the unique graph of minimum algebraic connectivity in the class of connected graphs $\begin{array}{} {\it\Omega}^c_{n}={\it\Omega}^c_{1,n}\cup{\it\Omega}^c_{2,n}, \end{array}$ where $\begin{array}{} {\it\Omega}^c_{1,n} \end{array}$ and $\begin{array}{} {\it\Omega}^c_{2,n} \end{array}$ are classes of the connected graphs in which the complement of each graph of order n is a bicyclic graph with exactly two and three cycles, respectively.

A NOTE ON GENERALIZED RAINBOW CONNECTION OF CONNECTED GRAPHS AND THEIR NUMBER OF EDGES

2021 ◽  
Vol 66 (3) ◽  
pp. 3-7
Author(s):  
Anh Nguyen Thi Thuy ◽  
Duyen Le Thi
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Connection Number ◽  
Connected Graphs ◽  
Rainbow Connection Number ◽  
Rainbow Connection ◽  
Rainbow Path ◽  
Vertex Disjoint

Let l ≥ 1, k ≥ 1 be two integers. Given an edge-coloured connected graph G. A path P in the graph G is called l-rainbow path if each subpath of length at most l + 1 is rainbow. The graph G is called (k, l)-rainbow connected if any two vertices in G are connected by at least k pairwise internally vertex-disjoint l-rainbow paths. The smallest number of colours needed in order to make G (k, l)-rainbow connected is called the (k, l)-rainbow connection number of G and denoted by rck,l(G). In this paper, we first focus to improve the upper bound of the (1, l)-rainbow connection number depending on the size of connected graphs. Using this result, we characterize all connected graphs having the large (1, 2)-rainbow connection number. Moreover, we also determine the (1, l)-rainbow connection number in a connected graph G containing a sequence of cut-edges.

scholarly journals Closed paths whose steps are roots of unity

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Gilbert Labelle ◽  
Annie Lacasse
Keyword(s):  
Roots Of Unity ◽  
Explicit Formulas ◽  
Asymptotic Expressions ◽  
International Audience ◽  
Dual Sequences ◽  
Polygonal Paths

International audience We give explicit formulas for the number $U_n(N)$ of closed polygonal paths of length $N$ (starting from the origin) whose steps are $n^{\textrm{th}}$ roots of unity, as well as asymptotic expressions for these numbers when $N \rightarrow \infty$. We also prove that the sequences $(U_n(N))_{N \geq 0}$ are $P$-recursive for each fixed $n \geq 1$ and leave open the problem of determining the values of $N$ for which the $\textit{dual}$ sequences $(U_n(N))_{n \geq 1}$ are $P$-recursive. Nous donnons des formules explicites pour le nombre $U_n(N)$ de chemins polygonaux fermés de longueur $N$ (débutant à l'origine) dont les pas sont des racines $n$-ièmes de l'unité, ainsi que des expressions asymptotiques pour ces nombres lorsque $N \rightarrow \infty$. Nous démontrons aussi que les suites $(U_n(N))_{N \geq 0}$ sont $P$-récursives pour chaque $n \geq 1$ fixé et laissons ouvert le problème de déterminer les valeurs de $N$ pour lesquelles les suites $\textit{duales}$ $(U_n(N))_{n \geq 1}$ sont $P$-récursives.

scholarly journals The Structure of Locally Finite Two-Connected Graphs

10.37236/1211 ◽  
1995 ◽  
Vol 2 (1) ◽  
Author(s):  
Carl Droms ◽  
Brigitte Servatius ◽  
Herman Servatius
Keyword(s):  
Connected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Locally Finite ◽  
Connected Graphs ◽  
Finite Graphs ◽  
Locally Finite Graphs ◽  
Necessary And Sufficient ◽  
Block Tree ◽  
Labeled Tree

We expand on Tutte's theory of $3$-blocks for $2$-connected graphs, generalizing it to apply to infinite, locally finite graphs, and giving necessary and sufficient conditions for a labeled tree to be the $3$-block tree of a $2$-connected graph.

scholarly journals On generalized 3-connectivity of the strong product of graphs

2018 ◽  
Vol 12 (2) ◽  
pp. 297-317
Author(s):  
Encarnación Abajo ◽  
Rocío Casablanca ◽  
Ana Diánez ◽  
Pedro García-Vázquez
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Sharp Bound ◽  
Connected Graphs ◽  
Strong Product ◽  
Generalized Connectivity ◽  
Internally Disjoint Trees ◽  
Product Of Graphs

Let G be a connected graph with n vertices and let k be an integer such that 2 ? k ? n. The generalized connectivity kk(G) of G is the greatest positive integer l for which G contains at least l internally disjoint trees connecting S for any set S ? V (G) of k vertices. We focus on the generalized connectivity of the strong product G1 _ G2 of connected graphs G1 and G2 with at least three vertices and girth at least five, and we prove the sharp bound k3(G1 _ G2) ? k3(G1)_3(G2) + k3(G1) + k3(G2)-1.

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