scholarly journals Sobolev Regularity of Multilinear Fractional Maximal Operators on Infinite Connected Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2883
Author(s):  
Suying Liu ◽  
Feng Liu
Keyword(s):  
Sobolev Spaces ◽  
Connected Graph ◽  
Maximal Operators ◽  
Connected Graphs ◽  
Sobolev Regularity ◽  
Geometric Conditions

Let G be an infinite connected graph. We introduce two kinds of multilinear fractional maximal operators on G. By assuming that the graph G satisfies certain geometric conditions, we establish the bounds for the above operators on the endpoint Sobolev spaces and Hajłasz–Sobolev spaces on 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 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.

Distance spectral radius of unicyclic graphs with fixed maximum degree

2019 ◽  
Vol 19 (04) ◽  
pp. 2050068
Author(s):  
Hezan Huang ◽  
Bo Zhou
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Maximum Degree ◽  
Distance Matrix ◽  
The Other ◽  
Connected Graphs ◽  
Largest Eigenvalue ◽  
Unicyclic Graphs ◽  
The Largest Eigenvalue

The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. For integers [Formula: see text] and [Formula: see text] with [Formula: see text], we prove that among the connected graphs on [Formula: see text] vertices of given maximum degree [Formula: see text] with at least one cycle, the graph [Formula: see text] uniquely maximizes the distance spectral radius, where [Formula: see text] is the graph obtained from the disjoint star on [Formula: see text] vertices and path on [Formula: see text] vertices by adding two edges, one connecting the star center with a path end, and the other being a chord of the star.

Some results about component factors in graphs

2019 ◽  
Vol 53 (3) ◽  
pp. 723-730 ◽  
Author(s):  
Sizhong Zhou
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Connected Graphs ◽  
Spanning Subgraph ◽  
Component Factor ◽  
Path Factor

For a set ℋ of connected graphs, a spanning subgraph H of a graph G is called an ℋ-factor of G if every component of H is isomorphic to a member ofℋ. An H-factor is also referred as a component factor. If each component of H is a star (resp. path), H is called a star (resp. path) factor. By a P≥ k-factor (k positive integer) we mean a path factor in which each component path has at least k vertices (i.e. it has length at least k − 1). A graph G is called a P≥ k-factor covered graph, if for each edge e of G, there is a P≥ k-factor covering e. In this paper, we prove that (1) a graph G has a {K1,1,K1,2, … ,K1,k}-factor if and only if bind(G) ≥ 1/k, where k ≥ 2 is an integer; (2) a connected graph G is a P≥ 2-factor covered graph if bind(G) > 2/3; (3) a connected graph G is a P≥ 3-factor covered graph if bind(G) ≥ 3/2. Furthermore, it is shown that the results in this paper are best possible in some sense.

New bounds on the independence number of connected graphs

2018 ◽  
Vol 10 (05) ◽  
pp. 1850069
Author(s):  
Nader Jafari Rad ◽  
Elahe Sharifi
Keyword(s):  
Lower Bound ◽  
Connected Graph ◽  
Maximum Degree ◽  
Applied Mathematics ◽  
Independence Number ◽  
Independent Set ◽  
Maximum Cardinality ◽  
Connected Graphs ◽  
Cut Vertex

The independence number of a graph [Formula: see text], denoted by [Formula: see text], is the maximum cardinality of an independent set of vertices in [Formula: see text]. [Henning and Löwenstein An improved lower bound on the independence number of a graph, Discrete Applied Mathematics  179 (2014) 120–128.] proved that if a connected graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] does not belong to a specific family of graphs, then [Formula: see text]. In this paper, we strengthen the above bound for connected graphs with maximum degree at least three that have a non-cut-vertex of maximum degree. We show that if a connected graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] has a non-cut-vertex of maximum degree then [Formula: see text], where [Formula: see text] is the maximum degree of the vertices of [Formula: see text]. We also characterize all connected graphs [Formula: see text] of order [Formula: see text] and size [Formula: see text] that have a non-cut-vertex of maximum degree and [Formula: see text].

5-Shredders of Contraction-Critical 5-Connected Graphs

2020 ◽  
Vol 30 (03) ◽  
pp. 2040008
Author(s):  
Chengfu Qin ◽  
Weihua Yang
Keyword(s):  
Connected Graph ◽  
Connected Graphs ◽  
Spanning Subgraph

Yoshimi Egawa [8] showed that a 5-connected graph G admits at most [Formula: see text] 5-shredders. In this paper we shown that a contraction-critical 5-connected graph G admits at most [Formula: see text] 5-shredders. Further we show that, for every contraction-critical 5-connected graph G, there is a contraction critical 5-connected graph [Formula: see text] such that G is a spanning subgraph of [Formula: see text] and [Formula: see text] admits at most [Formula: see text] 5-shredders.

scholarly journals The edge geodetic self decomposition number of a graph

2020 ◽  
Author(s):  
John J ◽  
Stalin D
Keyword(s):  
Connected Graph ◽  
Maximum Cardinality ◽  
Connected Graphs ◽  
Edge Disjoint ◽  
General Properties ◽  
Geodetic Number ◽  
Simple Connected Graph ◽  
Decomposition Number

Let  G = (V, E)  be a simple connected  graph  of order  p and  size q.  A decomposition  of a graph  G is a collection  π  of edge-disjoint sub graphs  G1, G2, ..., Gn  of G such  that every  edge of G belongs to exactly  one Gi , (1 ≤ i ≤ n) . The decomposition  π = {G1, G2, ....Gn } of a connected  graph  G is said to be an edge geodetic self decomposi- tion  if ge (Gi ) = ge (G), (1 ≤ i ≤ n).The maximum  cardinality of π is called the edge geodetic self decomposition  number of G and is denoted by πsge (G), where ge (G) is the edge geodetic number  of G.  Some general properties   satisfied  by  this  concept  are  studied.    Connected  graphs which are edge geodetic self decomposable  are characterized.

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