Dynamic interwell connectivity analysis of multi-layer waterflooding reservoirs based on an improved graph neural network

The analysis of interwell connectivity plays an important role in the formulation of oilfield development plans and the description of residual oil distribution. In fact, sandstone reservoirs in China's onshore oilfields generally have the characteristics of thin and many layers, so multi-layer joint production is usually adopted. It remains a challenge to ensure the accuracy of splitting and dynamic connectivity in each layer of the injection-production wells with limited field data. The three-dimensional well pattern of multi-layer reservoir and the relationship between injection-production wells can be equivalent to a directional heterogeneous graph. In this paper, an improved graph neural network is proposed to construct an interacting process mimics the real interwell flow regularity. In detail, this method is used to split injection and production rates by combining permeability, porosity and effective thickness, and to invert the dynamic connectivity in each layer of the injection-production wells by attention mechanism.Based on the material balance and physical information, the overall connectivity from the injection wells,through the water injection layers to the production layers and the output of final production wells is established. Meanwhile, the change of well pattern caused by perforation, plugging and switching of wells at different times is achieved by updated graph structure in spatial and temporal ways. The effectiveness of the method is verified by a combination of reservoir numerical simulation examples and field example. The method corresponds to the actual situation of the reservoir, has wide adaptability and low cost, has good practical value, and provides a reference for adjusting the injection-production relationship of the reservoir and the development of the remaining oil.

Cayley Picture Fuzzy Graphs and Interconnected Networks

Theory of the Cayley graphs is directly linked with the group theory.However,if there are uncertainties on the vertices or edges or both then fuzzy graphs have an extraordinary importance.In this perspective,numbers of generalηizations of fuzzy graphs have been explored in the literature.Among the others,picture fuzzy graph(PFG)has its own importance.A picture fuzzy graph(PFG)is a pair G=(C,D)defined on a H^(*)=(A,B),where C=(ηC,θ_(C),■_(C))is a picture fuzzy set on A and D=(ηD,θ_(D),■_(D))is a picture fuzzy set over the set B∈A×A such that for any edge mn∈ B with ηD(m,n)≤min(ηC(m),ηC(n)),θD(m,n)≤min(θC(m),θC(n))and ■_(D)(m,n)≥max(■_(C)(m),■_(C)(n)).In this manuscript,we introduce the notion of the Cayley picture fuzzy graphs on groups which is the generalization of the picture fuzzy graphs.Firstly,we discuss few important characteristics of the Cayley picture fuzzy graphs.We show that Cayley picture fuzzy graphs are vertex transitive and hence regular.Then,we investigate different types of Cayley graphs induced by the Cayley picture fuzzy graphs by using different types of cuts.We extensively discuss the term connectivity of the Cayley picture fuzzy graphs.Vertex connectivity and edge connectivity of the Cayley picture fuzzy graphs are also addressed.We also investigate the linkage between these two.Throughout,we provide the extensions of some characηteristics of both the PFGs and Cayley fuzzy graphs in the setting of Cayley picture fuzzy graphs.Finally,we provide the model of interconnected networks based on the Cayley picture fuzzy graphs.

A Note on the Inverse Connected p-Median Problem on Block Graphs

Recently, the inverse connected p-median problem on block graphs G(V,E,w) under various cost functions, say rectilinear norm, Chebyshev norm, and bottleneck Hamming distance. Their contributions include finding a necessary and sufficient condition for the connected p-median problem on block graphs, developing algorithms and showing that these problems can be solved in O(n log n) time, where n is the number of vertices in the underlying block graph. Using similar technique, we show that some results are incorrect by a counter-example. Then we redefine some notations, reprove Theorem 1 and redescribe Theorem 2, Theorem 3 and Theorem 4.

Regional differences in gully network connectivity based on graph theory:a case study on the Loess Plateau,China

Material exchange frequently occurs in gullies,and thus the relationship between a gullynetwork structure and sediment transport potential has attracted considerable interest.However,previous researches ignored the difficulty of material transport from sources to sinks,and did not quantify the connectivity of a network structure.In this study,we used a graph model structure to model gully networks of six typical sample areas in the Loess Plateau of China and quantified gully network connectivity using four indexes:average node strength,accessibility from sources to sinks,potential flow,and network structural connectivity index.Results show that:(1)Reflected by different quantitative indexes,the trends of gully network connectivity in different regions are similar.From north to south,the connectivity of a sample area first increases and then decreases.(2)The more mature gullies have stronger network connectivity.Small resistance is conducive to material transport in the gullies.(3)The node connectivity index of the gully network shows a significant aggregation distribution in space,and node connectivity on the main channel is often stronger than that on the branch trench.These results not only deepen the understanding of the process and mechanism of loess gully geomorphic development and evolution but also provide a reference for geomorphic studies.

A result on quasi k-connected graphs

Let G be a k-connected graph, and T be a subset of V（G）. If G-T is not connected,then T is said to be a cut-set of G. A k-cut-set T of G is a cut-set of G with │T│=k. Let T bea k-cut-set of a k-connected graph G. If G - T can be partitioned into subgraphs G1 and G2such that │G1│≥ 2, │G2│ 〉 2, then we call T a nontrivial k-cut-set of G. Suppose that G is a（k-1）-connected graph without nontrivial （k - 1）-cut-set. Then we call G a quasi k-connectedgraph. In this paper, we prove that for any integer k ≥ 5, if G is a k-connected graph withoutK4-, then every vertex of G is incident with an edge whose contraction yields a quasi k-connectedgraph, and so there are at least │V（G）│/2 edges of G such that the contraction of every member ofthem results in a quasi k-connected graph.

On the numbers of connected components of hattorigraph

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Depth First:Optimal Path Discovery Between Designated Nodes in Random Ring-Based Graphs

This paper focuses on optimally determining the existence of connected paths between some given nodes in random ring-based graphs.Serving as a fundamental underlying structure in network modeling,ring topology appears as commonplace in many realistic scenarios.Regarding this,we consider graphs composed of rings,with some possible connected paths between them.Without prior knowledge of the exact node permutations on rings,the existence of each edge can be unraveled through edge testing at a unit cost in one step.The problem examined is that of determining whether the given nodes are connected by a path or separated by a cut,with the minimum expected costs involved.Dividing the problem into different cases based on different topologies of the ring-based networks,we propose the corresponding policies that aim to quickly seek the paths between nodes.A common feature shared by all those policies is that we stick to going in the same direction during edge searching,with edge testing in each step only involving the test between the source and the node that has been tested most.The simple searching rule,interestingly,can be interpreted as a delightful property stemming from the neat structure of ring-based networks,which makes the searching process not rely on any sophisticated behaviors.We prove the optimality of the proposed policies by calculating the expected cost incurred and making a comparison with the other class of strategies.The effectiveness of the proposed policies is also verified through extensive simulations,from which we even disclose three extra intriguing findings:i)in a onering network,the cost will grow drastically with the number of designated nodes when the number is small and will grow slightly when that number is large;ii)in ring-based network,Depth First is optimal in detecting the connectivity between designated nodes;iii)the problem of multi-ring networks shares large similarity with that of two-ring networks,and a larger number of ties between rings will not influence the expected cost.

Structural and functional connectivity of the whole brain and subnetworks in individuals with mild traumatic brain injury:predictors of patient prognosis

Patients with mild traumatic brain injury have a diverse clinical presentation,and the underlying pathophysiology remains poorly understood.Magnetic resonance imaging is a non-invasive technique that has been widely utilized to investigate neuro biological markers after mild traumatic brain injury.This approach has emerged as a promising tool for investigating the pathogenesis of mild traumatic brain injury.G raph theory is a quantitative method of analyzing complex networks that has been widely used to study changes in brain structure and function.However,most previous mild traumatic brain injury studies using graph theory have focused on specific populations,with limited exploration of simultaneous abnormalities in structural and functional connectivity.Given that mild traumatic brain injury is the most common type of traumatic brain injury encounte red in clinical practice,further investigation of the patient characteristics and evolution of structural and functional connectivity is critical.In the present study,we explored whether abnormal structural and functional connectivity in the acute phase could serve as indicators of longitudinal changes in imaging data and cognitive function in patients with mild traumatic brain injury.In this longitudinal study,we enrolled 46 patients with mild traumatic brain injury who were assessed within 2 wee ks of injury,as well as 36 healthy controls.Resting-state functional magnetic resonance imaging and diffusion-weighted imaging data were acquired for graph theoretical network analysis.In the acute phase,patients with mild traumatic brain injury demonstrated reduced structural connectivity in the dorsal attention network.More than 3 months of followup data revealed signs of recovery in structural and functional connectivity,as well as cognitive function,in 22 out of the 46 patients.Furthermore,better cognitive function was associated with more efficient networks.Finally,our data indicated that small-worldness in the acute stage could serve as a predictor of longitudinal changes in connectivity in patients with mild traumatic brain injury.These findings highlight the importance of integrating structural and functional connectivity in unde rstanding the occurrence and evolution of mild traumatic brain injury.Additionally,exploratory analysis based on subnetworks could serve a predictive function in the prognosis of patients with mild traumatic brain injury.

A Sufficient Condition of Hamilton Connected Graph

Let G be a 3-connected graph with n vertices. The paper proves that if for each pair of vertices u and v of G, d(u,v)=2, has ｜N(u)∩N(v)｜≤α(α is the minimum independent set number), and then max{d(u),d(v)}≥n+12, then G is a Hamilton connected graph.

Note on Faudree-Schelp Theorem in Path Connected Graphs and Ore Theorem in Hamilton Connected Graphs

In this note more short proofs are given for Faudree-Schelp theorem and Ore theorem.

On Graphs with Equal Connected Domination and 2-connected Domination Numbers

A subset S of V is called a k-connected dominating set if S is a dominating set and the induced subgraph S has at most k components.The k-connected domination number γck（G） of G is the minimum cardinality taken over all minimal k-connected dominating sets of G.In this paper,we characterize trees and unicyclic graphs with equal connected domination and 2-connected domination numbers.

An Alternative Proof of the Largest Number of Maximal Independent Sets in Connected Graphs Having at Most Two Cycles

G. C. Ying, Y. Y. Meng, B. E. Sagan, and V. R. Vatter [1] found the maximum number of maximal independent sets in connected graphs which contain at most two cycles. In this paper, we give an alternative proof to determine the largest number of maximal independent sets among all connected graphs of order n ≥ 12, which contain at most two cycles. We also characterize the extremal graph achieving this maximum value.

A TIGHT LOWER BOUND ON THE MAXIMUM GENUS OF A 3-CONNECTED LOOPLESS MULTIGRAPH

It is proved that every 3 connected loopless multigraph has maximum genus at least one third of its cycle rank plus one if its cycle rank is not less than ten, and if its cycle rank is less than ten,it is upper embeddable.This lower bound is tight.There are infinitely many 3 connected loopless multigraphs attaining this bound.

Super Cyclically Edge Connected Half Vertex Transitive Graphs

Tian and Meng in [Y. Tian and J. Meng, λc -Optimally half vertex transitive graphs with regularity k, Information Processing Letters 109 (2009) 683 - 686] shown that a connected half vertex transitive graph with regularity k and girth g(G) ≥ 6 is cyclically optimal. In this paper, we show that a connected half vertex transitive graph G is super cyclically edge-connected if minimum degree δ(G) ≥ 6 and girth g(G) ≥ 6.

ON THE EXISTENCE OF THE h-RESTRICTED EDGE CONNECTIVITY OF A GRAPH

Let h be a nonnegative integer. The h-restricted edge connectivity λ h(G) of a simple connected graph G is defined as the minimum cardinality over the sets of edges of G, if any, whose removal disconnects G and every component of the resulting graph has more than h vertices. This paper gave a necessary and sufficient condition and also three useful sufficient conditions to guarantee the existence of λ h(G). Moreover, it explicitly characterized the graphs whose 2-restricted edge connectivities do not exist.

The atom-bond connectivity index of chemical bicyclic graphs

The atom-bond connectivity（ABC） index provides a good model for the stability of linear and branched alkanes as well as the strain energy of cycloalkanes,which is defined as ABC（G） =∑ uv∈E（G） √d u＋dv-2 dudv,where du denotes the degree of a vertex u in G.A chemical graph is a graph in which no vertex has degree greater than 4.In this paper,we obtain the sharp upper and lower bounds on ABC index of chemical bicyclic graphs.

SOME RESULTSON R_2-EDGE-CONNECTIVITY OF EVEN REGULAR GRAPHS

Let Gbe a connected k(≥3)-regulargraph w ith girth g. A setSofthe edgesin G is called an R2-edge-cutifG- Sis disconnected and contains neither an isolated vertex nor a one- degree vertex. The R2-edge-connectivity of G, denoted by λ″(G), is the m inim um cardinality over allR2-edge-cuts, w hich is an im portantm easure for fault-tolerance of com puter intercon- nection netw orks. In this paper, λ″(G)= g(2k- 2) for any 2k-regular connected graph G(≠ K5) that is either edge-transitive or vertex-transitive and g≥5 is given.

Restricted Edge Connectivity of Binary Undirected Kautz Graphs

A restricted edge cut is an edge cut of a connected graph whose removal resultsin a disconnected graph without isolated vertices. The size of a minimum restricted edge cutof a graph G is called its restricted edge connectivity, and is denoted by λ′(G). Let ξ(G) bethe minimum edge degree of graph G. It is known that λ′(G) ≤ξ(G) if G contains restrictededge cuts. Graph G is called maximal restricted edge connected if the equality holds in thethe preceding inequality. In this paper, undirected Kautz graph UK(2, n) is proved to bemaximal restricted edge connected if n ≥ 2.

On Eccentric Connectivity Index and Polynomial of Thorn Graph

The eccentric connectivity index based on degree and eccentricity of the vertices of a graph is a widely used graph invariant in mathematics. In this paper we present the explicit generalized expressions for the eccentric connectivity index and polynomial of the thorn graphs, and then consider some particular cases.

Hamilton Graphs Involving Connectivity

Let G be a 2 connected simple graph of order n and connectivity k .Bauer, Broersma and Li proved that for an independent set S=u,v,w, d(u)+d(v)+d(w)≥n+k ,then G is Hamiltonian. This paper improves the result.Let S be an independent set. If there exist u,v∈S,du,v=2, then S is called a 2 independent set. This paper proves the following result. Let G be a simple graph of order n and connectivity k≥2 . If for every 2 independent set S=u,v,w, d(u)+d(v)+d(w)≥n+k , then G is Hamiltonian. This result implies that we may consider all triples of 2 independent set instead of all triples of independent set.