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Marine transportation is pivotal in the rapid development of global trade, significantly enhancing international economic and trade connectivity and impacting the sustainable development of the global economy. In this study, we developed a novel technical framework based on the Laplacian matrix to evaluate the node significance and edge connectivity of the global shipping network using 2019 vessel schedule data from the top 30 liner shipping companies, as ranked by Alphaliner. Our analyses were conducted in both L-space, characterizing the connectivity function, and P-space, characterizing the transfer function. The findings indicate the following. (1) There is no consistent relationship between node significance and centrality for most ports. Ports with high node significance are mainly located in the Asia–Pacific region, with Singapore Port being the port with the highest node significance in L-space and Shanghai Port being the port with the highest node significance in P-space. (2) In L-space, the structures with significant improvements in edge connectivity in the shipping network have at least one port node that exhibits both low degree centrality and low betweenness centrality; these are primarily found on East African routes. (3) In P-space, the structures with significant improvements in edge connectivity in the shipping network are more complex but are notably linked to the ports of Assaluyeh and Bandar Abbas in Iran. The proposed node evaluation and edge addition strategy effectively analyze port significance and edge connectivity, providing decision-making support for optimizing port layouts, supporting container route planning, and enhancing the overall performance of the shipping network. [ABSTRACT FROM AUTHOR]

Adolescence is a crucial period for physical and psychological development. The impact of negative life events represents a risk factor for the onset of neuropsychiatric disorders. This study aims to investigate the relationship between negative life events and structural brain connectivity, considering both graph theory and connectivity strength. A group (n = 487) of adolescents from the IMAGEN Consortium was divided into Low and High Stress groups. Brain networks were extracted at an individual level, based on morphological similarity between grey matter regions with regions defined using an atlas-based region of interest (ROI) approach. Between-group comparisons were performed with global and local graph theory measures in a range of sparsity levels. The analysis was also performed in a larger sample of adolescents (n = 976) to examine linear correlations between stress level and network measures. Connectivity strength differences were investigated with network-based statistics. Negative life events were not found to be a factor influencing global network measures at any sparsity level. At local network level, between-group differences were found in centrality measures of the left somato-motor network (a decrease of betweenness centrality was seen at sparsity 5%), of the bilateral central visual and the left dorsal attention network (increase of degree at sparsity 10% at sparsity 30% respectively). Network-based statistics analysis showed an increase in connectivity strength in the High stress group in edges connecting the dorsal attention, limbic and salience networks. This study suggests negative life events alone do not alter structural connectivity globally, but they are associated to connectivity properties in areas involved in emotion and attention.

[M. Ma, The connectivity of exchanged hypercubes, Discrete Math., Algorithms Appl. 2(2) (2010) 213–220.] proved that the vertex connectivity and the edge connectivity of the exchanged hypercube are each equal to the minimum degree of the graph. This paper presents a proof of that result, which is substantially simpler and more intuitive. [ABSTRACT FROM AUTHOR]

The problem of efficiently computing all k -edge-connected components (k -ECCs) of a graph G for a user-givenk has been extensively studied recently in view of its importance in many applications. The k -ECCs of G for all possible values ofk form a hierarchical structure; that is, any two different k -ECCs for the same k value are disjoint and any k -ECC is contained in a unique (k - 1) -ECC. In this paper, we study the problem of efficiently constructing the hierarchy tree of the k -ECCs for all possible k values, for a graph G. The existing approaches TD and BU construct the hierarchy tree in either a top-down manner or a bottom-up manner, with both having the time complexity of O (δ (G) × T KECC (G)) , where δ (G) is the degeneracy of G and T KECC (G) is the time complexity of computing all k -ECCs of G for a specific k value. Here, the degeneracy of G is defined as the maximum value among the minimum vertex degrees of all subgraphs of G and is at most m where m is the number of edges in G. To improve the time complexity, we propose a divide-and-conquer approach DC running in O ((log δ (G)) × T KECC (G)) time; this time complexity is optimal up to a logarithmic factor. However, a straightforward implementation of DC would take O ((m + n) log δ (G)) main-memory space, which could easily run out-of-memory when processing large graphs; here, n is the number of vertices in G. To reduce the main-memory footprint of our algorithm, we propose adjacency array-based techniques to optimize the space complexity to 2 m + O (n log δ (G)) and denote our resulting algorithm by DC - AA . As a by-product of DC - AA , we also improve the space complexity of the state-of-the-art algorithm for computing all k -ECCs for a specific k to 2 m + O (n) , by using the same technique as used in DC - AA . Finally, we propose optimization techniques to improve the practical efficiency of the existing approach BU and denote the space-optimized version of it as BU ∗ - AA which runs in O (δ (G) × T KECC (G)) time and 2 m + O (n) space. Extensive experiments on large real graphs and synthetic graphs demonstrate that our algorithms DC - AA and BU ∗ - AA outperform the state-of-the-art approaches by up to 28 times in terms of running time and by up to 8 times in terms of main memory usage. In particular, our approach BU ∗ - AA processes the Twitter graph, which has more than 1 billion undirected edges, in 29 min with 13.5 GB memory, while the state-of-the-art approaches take more than 13 h after our space optimization; note that the state-of-the-art approaches run out-of-memory if without our space optimization. Our empirical study also shows that BU ∗ - AA , despite having a higher time complexity, performs better than DC - AA in practice. We also remark that BU ∗ - AA is much simpler and easier to implement than DC - AA . [ABSTRACT FROM AUTHOR]

The spanning tree packing number of a graph G $G$, denoted by τ(G) $\tau (G)$, is the maximum number of edge‐disjoint spanning trees contained in G $G$. The study of τ(G) $\tau (G)$ is one of the classic problems in graph theory. Cioabă and Wong initiated to investigate τ(G) $\tau (G)$ from spectral perspectives in 2012 and since then, τ(G) $\tau (G)$ has been well studied using the second‐largest eigenvalue of the adjacency matrix in the past decade. In this paper, we further extend the results in terms of the number of edges and the spectral radius, respectively; and prove tight sufficient conditions to guarantee τ(G)≥k $\tau (G)\ge k$ with extremal graphs characterized. Moreover, we confirm a conjecture of Ning, Lu, and Wang on characterizing graphs with the maximum spectral radius among all graphs with a given order as well as fixed minimum degree and fixed edge connectivity. Our results have important applications in rigidity and nowhere‐zero flows. We conclude with some open problems in the end. [ABSTRACT FROM AUTHOR]

A graph G is said to be a semi-splitting block graph if there exists a graph H such that SB(H) ≌ G. This paper establishes a characterisation of semi-splitting block graphs based on the partition of the vertex set of G. The vertex (edge) connectivity and p-connectedness (p-edge connectedness) of SB(G) are examined. For all integers a, b with 1 < a < b, the existence of the graph G for which κ (G) = a, κ (SB(G)) = b and λ (G) = a, λ (SB(G)) = b are proved independently. The characterization of graphs with κ(SB(G)) = κ (G) and a necessary condition for graphs with κ (SB(G)) = λ (SB(G)) are achieved.

Marine transportation is pivotal in the rapid development of global trade, significantly enhancing international economic and trade connectivity and impacting the sustainable development of the global economy. In this study, we developed a novel technical framework based on the Laplacian matrix to evaluate the node significance and edge connectivity of the global shipping network using 2019 vessel schedule data from the top 30 liner shipping companies, as ranked by Alphaliner. Our analyses were conducted in both L-space, characterizing the connectivity function, and P-space, characterizing the transfer function. The findings indicate the following. (1) There is no consistent relationship between node significance and centrality for most ports. Ports with high node significance are mainly located in the Asia–Pacific region, with Singapore Port being the port with the highest node significance in L-space and Shanghai Port being the port with the highest node significance in P-space. (2) In L-space, the structures with significant improvements in edge connectivity in the shipping network have at least one port node that exhibits both low degree centrality and low betweenness centrality; these are primarily found on East African routes. (3) In P-space, the structures with significant improvements in edge connectivity in the shipping network are more complex but are notably linked to the ports of Assaluyeh and Bandar Abbas in Iran. The proposed node evaluation and edge addition strategy effectively analyze port significance and edge connectivity, providing decision-making support for optimizing port layouts, supporting container route planning, and enhancing the overall performance of the shipping network.

Edge detection is a fundamental technique in image processing and computer vision that plays a crucial role in various applications including object recognition, shape analysis, segmentation, feature extraction, image enhancement, image understanding, compression, and preprocessing. It involves identifying the boundaries or edges between different objects or regions which enables us to extract valuable visual information, analyze image structure, and facilitate subsequent computer vision tasks. In this study, we highlighted two primary issues that are faced in edge detection: edge connectivity and edge thickness. Choosing the right threshold for reliable edge detection has long been one of the most difficult problems in traditional edge detection techniques. To deal with these issues, the present research is motivated to execute a new deep learning-based edge detection model named MultiResEdge which is a tweaked variant of the original UNet framework. Instead of directly integrating encoder and decoder features, the suggested model makes use of a MutiRes block to modify spatial information from different scales, and a residual path with a set of convolutions to convey the features from the encoder to the decoder. To reduce the gap between the encoder and decoder features, we advocate using a combination of convolutional and skip connections. We also incorporated semantic edge information with semantic curves to improve the border. In addition, we have integrated a multi-step preprocessing technique that mitigates the false positives and false negatives and hence improves the accuracy, precision, recall, and F1 score. The extensive experiment is performed using images from two benchmark datasets to justify the predicted edge comparing it with the corresponding ground truth in terms of various objective metrics (such as entropy, MSE, PSNR, SSIM, and FSIM). Overall, our proposed edge detection model has obtained an accuracy of 99% along with an F1 score of 98% which is better than the related state-of-the-art methodologies.

The Sombor index is one of the geometry-based descriptors, which was defined as S O (G) = ∑ u v ∈ E (G) d u 2 + d v 2 , where d u (resp. d v ) denotes the degree of vertex u (resp. v) in G. In this note, we determine the graphs among the set of graphs with vertex connectivity (resp. edge connectivity) at most k having the maximum and minimum Sombor indices, 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 conclusions of the above paper, we first give some counterexamples, then provide another simple proof about the minimum Sombor indices of graphs with n vertices, k cut vertices and at least one cycle. [ABSTRACT FROM AUTHOR]

A multigraph is a graph with possible multiple edges, but no loops. Let t be a positive integer. Let G t be the set of simple graphs (or multigraphs) such that for each G ∈ G t there exist at least t + 1 non-empty disjoint proper subsets V 1 , V 2 , ... , V t + 1 ⊆ V (G) satisfying V (G) ∖ (V 1 ∪ V 2 ∪ ⋯ ∪ V t + 1) ≠ ϕ and edge connectivity κ ′ (G) = e (V i , V (G) ∖ V i) for i = 1 , 2 , ... , t + 1. Let D (G) and A (G) denote the degree diagonal matrix and adjacency matrix of a simple graph (or a multigraph) G , and let μ i (G) be the i th largest eigenvalue of the Laplacian matrix L (G) = D (G) + A (G). In this paper, we investigate the relationship between μ n − 2 (G) and edge connectivity or spanning tree packing number of a graph G ∈ G 1 , respectively. We also give the relationship between μ n − 3 (G) and edge connectivity or spanning tree packing number of a graph G ∈ G 2 , respectively. Moreover, we generalize all the results about L (G) to a more general matrix a D (G) + A (G) , where a is a real number with a ≥ − 1. [ABSTRACT FROM AUTHOR]

By the vertex connectivity of a graph we mean the smallest number of vertices whose removal results in a disconnected or trivial graph, i.e., a graph with one vertex. By the edge connectivity of a nontrivial graph we mean the smallest number of edges whose removal results in a disconnected graph. The vertex connectivity , the edge connectivity , and the minimum vertex degree of a graph are related by the following inequality found by Whitney in 1932: . More recently, Chartrand and Harary proved that for any positive integers , , and such that there exists a graph such that , , and . In the proof, one constructs a family of such graphs with vertices and edges each. However, it is easily seen that one can sometimes construct a suitable graph with fewer vertices and edges. For example, if , then the complete graph with vertices is optimal. In the present paper, we consider the problem of minimizing the Chartrand–Harary construction. We show that for , it is possible to construct a graph with fewer vertices. Four cases are considered separately: , , , and . For each case, we present graphs with the prescribed characteristics , , and and with minimum numbers of vertices and edges. [ABSTRACT FROM AUTHOR]

We present a deterministic algorithm that computes the edge-connectivity of a graph in near-linear time. This is for a simple undirected unweighted graph G with n vertices and m edges. This is the first o(mn) time deterministic algorithm for the problem. Our algorithm is easily extended to find a concrete minimum edge-cut. In fact, we can construct the classic cactus representation of all minimum cuts in near-linear time. The previous fastest deterministic algorithm by Gabow from STOC '91 took Õ(m+λ2 n), where λ is the edge connectivity, but λ can be as big as n−1. Karger presented a randomized near-linear time Monte Carlo algorithm for the minimum cut problem at STOC'96, but the returned cut is only minimum with high probability. Our main technical contribution is a near-linear time algorithm that contracts vertex sets of a simple input graph G with minimum degree Δ, producing a multigraph Ḡ with Õ(m/Δ) edges, which preserves all minimum cuts of G with at least two vertices on each side. In our deterministic near-linear time algorithm, we will decompose the problem via low-conductance cuts found using PageRank a la Brin and Page (1998), as analyzed by Andersson, Chung, and Lang at FOCS'06. Normally, such algorithms for low-conductance cuts are randomized Monte Carlo algorithms, because they rely on guessing a good start vertex. However, in our case, we have so much structure that no guessing is needed. [ABSTRACT FROM AUTHOR]

We initiate the study of k-edge-connected orientations of undirected graphs through edge flips for k ≥ 2. We prove that in every orientation of an undirected 2k-edge-connected graph, there exists a sequence of edges such that flipping their directions one by one does not decrease the edge connectivity, and the final orientation is k-edge connected. This yields an "edge-flip based" new proof of Nash-Williams' theorem: A undirected graph G has a k-edge-connected orientation if and only if G is 2k-edge connected. As another consequence of the theorem, we prove that the edge-flip graph of k-edge-connected orientations of an undirected graph G is connected if G is (2k+2)-edge connected. This has been known to be true only when k=1. [ABSTRACT FROM AUTHOR]

In this research study, we first define the strong degree of a vertex in an m-polar fuzzy graph. Then we present various useful properties and prove some results concerning this new concept, in the case of complete m-polar fuzzy graphs. Further, we introduce the concept of m-polar fuzzy strength sequence of vertices, and we also investigate it in the particular instance of complete m-polar fuzzy graphs. We discuss connectivity parameters in m-polar fuzzy graphs with precise examples, and we investigate the m-polar fuzzy analogue of Whitney's theorem. Furthermore, we present a clustering method for vertices in an m-polar fuzzy graph based on the strength of connectedness between pairs of vertices. In order to formulate this method, we introduce terminologies such as ϵA-reachable vertices in m-polar fuzzy graphs, ϵA-connected m-polar fuzzy graphs, or ϵA-connected m-polar fuzzy subgraphs (in case the m-polar fuzzy graph itself is not ϵA-connected). Moreover, we discuss an application for clustering different companies in consideration of their multi-polar uncertain information. We then provide an algorithm to clearly understand the clustering methodology that we use in our application. Finally, we present a comparative analysis of our research work with existing techniques to prove its applicability and effectiveness.

Topological indices are important descriptors which can be used to characterize the structural properties of organic molecules from different aspects. The variable sum exdeg index $SEI_{a}(G)$ of a graph $G$ is defined as $\sum _{u\in V(G)}d_{G}(u)a^{d_{G}(u)}$, where $d_{G}(u)$ is the degree of vertex $u$ and $a$ is an arbitrary positive real number different from 1. In this paper, we obtain the extremal values of the variable sum exdeg indices (for $a>1$) in terms of the number of cut edges, or the number of cut vertices, or the vertex connectivity, or the edge connectivity of a graph. Furthermore, the corresponding extremal graphs are characterized.

Edge connectivity is often used to capture edge fault tolerance. As for most common networks, their edge connectivity is exactly equal to their minimum degree. Edge matroidal connectivity and conditional edge matroidal connectivity are two new graph edge connectivity parameters that can be defined when a partition of the edge set is given, for example when the network is built with different kinds of edges having different expected faultiness. And those two edge connectivity parameters can measure edge fault tolerance more than the traditional definition. The edge matroidal connectivity is based on the associated edge partition and the other is a generalization called conditional edge matroidal connectivity. We analyze these new parameters on hypercubes and folded hypercubes which are well studied networks. In this study, we consider their standard dimensional partition of the edges. This study leads to more structural insights about edge connectivity and yields many interesting questions. • We propose two new edge connectivity parameters for a graph based on the associated edge partition. • The two edge connectivity parameters are edge matroidal connectivity and conditional edge matroidal connectivity. • We analyze the (conditional) edge matroidal connectivity for hypercubes and folded hypercubes. [ABSTRACT FROM AUTHOR]

Both cortical and subcortical regions can be functionally organized into networks. Regions of the basal ganglia are extensively interconnected with the cortex via reciprocal connections that relay and modulate cortical function. Here we employ an edge-centric approach, which computes co-fluctuations among region pairs in a network to investigate the role and interaction of subcortical regions with cortical systems. By clustering edges into communities, we show that cortical systems and subcortical regions couple via multiple edge communities, with hippocampus and amygdala having a distinct pattern from striatum and thalamus. We show that the edge community structure of cortical networks is highly similar to one obtained from cortical nodes when the subcortex is present in the network. Additionally, we show that the edge community profile of both cortical and subcortical nodes can be estimates solely from cortico-subcortical interactions. Finally, we used a motif analysis focusing on edge community triads where a subcortical region coupled to two cortical regions and found that two community triads where one community couples the subcortex to the cortex were overrepresented. In summary, our results show organized coupling of the subcortex to the cortex that may play a role in cortical organization of primary sensorimotor/attention and heteromodal systems and puts forth the motif analysis of edge community triads as a promising method for investigation of communication patterns in networks.

In this paper, we consider the concept of the average edge-connectivity κ ′ ¯ (G) of a graph G , defined to be the average, over all pairs of vertices, of the maximum number of edge-disjoint paths connecting these vertices. Kim and O previously proved that κ ′ ¯ (G) n 2 ≥ n 2 + 7 n + 5 8 4 for any connected cubic graph on n ≥ 6 vertices. We refine their result by showing that κ ′ ¯ (G) n 2 ≥ 6 8   if n = 8 , 1 1 0     if n = 1 2 , n 2 + 7 n + 8 0 4 if n = 4 p , p ≥ 4 n 2 + 7 n + 5 8 4 if n = 4 p + 2 , p ≥ 2. We also characterize the graphs where equality holds. [ABSTRACT FROM AUTHOR]

The universal 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 universal 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 ≥ 4 is sufficiently small, then the universal cyclic edge-connectivity is (d - 2) g , providing a spectral condition for when this upper bound on universal cyclic edge-connectivity is tight. [ABSTRACT FROM AUTHOR]

Let G be a connected edge-weighted graph of order n and size m. Let w : E(G) → ℝ≥0 be the weighting function. We assume that w is normalised; that is, ∑e∈E(G) w(e) = m. The weighted distance dw(u, v) between any two vertices u and v is the least weight between them and the eccentricity ew(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, denoted as 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, m or edge-connectivity λ for two cases: G is a tree and G is connected but not a tree. In addition, we obtain the NordhausGaddum-type results for edge-weighted average eccentricity. [ABSTRACT FROM AUTHOR]

Plesnik proved that the edge connectivity and minimum degree are equal for diameter 2 graphs. We provide a streamlined proof of this fact and characterize the diameter 2 graphs with a nontrivial minimum edge cut.

An edge detection is important for its reliability and security which delivers a better understanding of object recognition in the applications of computer vision, such as pedestrian detection, face detection, and video surveillance. This paper introduced two fundamental limitations encountered in edge detection: edge connectivity and edge thickness, those have been used by various developments in the state-of-the-art. An optimal selection of the threshold for effectual edge detection has constantly been a key challenge in computer vision. Therefore, a robust edge detection algorithm using multiple threshold approaches (B-Edge) is proposed to cover both the limitations. The majorly used canny edge operator focuses on two thresholds selections and still witnesses a few gaps for optimal results. To handle the loopholes of the canny edge operator, our method selects the simulated triple thresholds that target to the prime issues of the edge detection: image contrast, effective edge pixels selection, errors handling, and similarity to the ground truth. The qualitative and quantitative experimental evaluations demonstrate that our edge detection method outperforms competing algorithms for mentioned issues. The proposed approach endeavors an improvement for both grayscale and colored images.

The power graph 𝒫 (G) of a finite group G is the undirected simple graph whose vertex set is G , in which two distinct vertices are adjacent if one of them is an integral power of the other. For an integer n ≥ 2 , let C n denote the cyclic group of order n and let r be the number of distinct prime divisors of n. The minimum degree δ (𝒫 (C n)) of 𝒫 (C n) is known for r ∈ { 1 , 2 } , see [R. P. Panda and K. V. Krishna, On the minimum degree, edge-connectivity and connectivity of power graphs of finite groups, Comm. Algebra 46(7) (2018) 3182–3197]. For r ≥ 3 , under certain conditions involving the prime divisors of n , we identify at most r − 1 vertices such that δ (𝒫 (C n)) is equal to the degree of at least one of these vertices. If r = 3 , or that n is a product of distinct primes, we are able to identify two such vertices without any condition on the prime divisors of n. [ABSTRACT FROM AUTHOR]

Adolescence is a crucial period for physical and psychological development. The impact of negative life events represents a risk factor for the onset of neuropsychiatric disorders. This study aims to investigate the relationship between negative life events and structural brain connectivity, considering both graph theory and connectivity strength. A group (n = 487) of adolescents from the IMAGEN Consortium was divided into Low and High Stress groups. Brain networks were extracted at an individual level, based on morphological similarity between grey matter regions with regions defined using an atlas-based region of interest (ROI) approach. Between-group comparisons were performed with global and local graph theory measures in a range of sparsity levels. The analysis was also performed in a larger sample of adolescents (n = 976) to examine linear correlations between stress level and network measures. Connectivity strength differences were investigated with network-based statistics. Negative life events were not found to be a factor influencing global network measures at any sparsity level. At local network level, between-group differences were found in centrality measures of the left somato-motor network (a decrease of betweenness centrality was seen at sparsity 5%), of the bilateral central visual and the left dorsal attention network (increase of degree at sparsity 10% at sparsity 30% respectively). Network-based statistics analysis showed an increase in connectivity strength in the High stress group in edges connecting the dorsal attention, limbic and salience networks. This study suggests negative life events alone do not alter structural connectivity globally, but they are associated to connectivity properties in areas involved in emotion and attention., Competing Interests: Dr. Banaschewski has served as an advisor or consultant to Actelion, Hexal Pharma,Bristol-Myers Squibb, Desitin Arzneimittel, Eli Lilly, Lundbeck, Medice, Neurim Pharmaceuticals, Novartis, Pfizer, and Shire, UCB, and Vifor Pharma; he has received conference attendance support, conference support, or speaking fees from Eli Lilly, Janssen McNeil, Medice, Novartis, and Shire, and UCB; and he is involved in clinical trials conducted by Eli Lilly, Novartis, and Shire and Viforpharma; he received royalities from Hogrefe, Kohlhammer, CIP Medien, Oxford University Press; the present work is unrelated to these relationships. Dr. Barker has received honoraria from General Electric Healthcare for teaching on scanner programming courses and acts as a consultant for IXICO. Dr. Poustka served in an advisory or consultancy role for Roche and Viforpharm and received speaker’s fee by Shire. She received royalties from Hogrefe, Kohlhammer and Schattauer. The present work is unrelated to the above grants and relationships. The other authors report no biomedical financial interests or potential conflicts of interest., (© 2024 The Authors.)

Let G be the set of simple graphs (or multigraphs) G such that for each G ∈ G there exists at least two non-empty disjoint proper subsets V 1 , V 2 ⊆ V (G) satisfying V (G) ∖ (V 1 ∪ V 2) ≠ φ and edge connectivity κ ′ (G) = e (V i , V (G) ∖ V i) for 1 ≤ i ≤ 2. A multigraph is a graph with possible multiple edges, but no loops. Let τ (G) be the maximum number of edge-disjoint spanning trees of a graph G. Motivated by a question of Seymour on the relationship between eigenvalues of a graph G and bounds of τ (G) , we mainly give the relationship between the third largest (signless Laplacian) eigenvalue and the bounds of κ ′ (G) and τ (G) of a simple graph or a multigraph G ∈ G , respectively. [ABSTRACT FROM AUTHOR]

Let k , n , s , t > 0 be integers and n = s + t ≥ 2 k + 2 . A simple bipartite graph G spanning K s , t is bi-k-maximal, if every subgraph of G has edge-connectivity at most k but any edge addition that does not break its bipartiteness creates a subgraph with connectivity at least k + 1 . We investigate the optimal size bounds of the bi-k-maximal simple graphs, and prove that if G is a bi-k-maximal graph with min { s , t } ≥ k on n vertices, then each of the following holds. Let m be an integer. Then there exists a bi-k-maximal graph G with m = | E (G) | if and only if m = n k - r k 2 + (r - 1) k for some integer r with 1 ≤ r ≤ ⌊ n 2 k + 2 ⌋ . Every bi-k-maximal graph G on n vertices satisfies | E (G) | ≤ (n - k) k , and this upper bound is tight. Every bi-k-maximal graph G on n vertices satisfies | E (G) | ≥ k (n - 1) - (k 2 - k) ⌊ n 2 k + 2 ⌋ , and this lower bound is tight. Moreover, the bi-k-maximal graphs reaching the optimal bounds are characterized. [ABSTRACT FROM AUTHOR]

Let G be a simple undirected connected graph. The signless Laplacian Estrada, Laplacian Estrada and Estrada indices of a graph G is the sum of the exponentials of the signless Laplacian eigenvalues, Laplacian eigenvalues and eigenvalues of G , respectively. The present work derives an upper bound for the Estrada index of a graph as a function of its chromatic number, in the family of graphs whose color classes have order not less than a fixed positive integer. The graphs for which the upper bound is tight is obtained. Additionally, an upper bound for the Estrada Index of the complement of a graph in the previous family of graphs with two color classes is given. A Nordhaus-Gaddum type inequality for the Laplacian Estrada index when G is a bipartite graph with color classes of order not less than 2, is presented. Moreover, a sharp upper bound for the Estrada index of the line graph and for the signless Laplacian index of a graph in terms of connectivity is obtained. [ABSTRACT FROM AUTHOR]

We give formulas, in terms of graph theoretical invariants, for the minimum distance and the generalized Hamming weights of the linear code generated by the rows of the incidence matrix of a signed graph over a finite field, and for those of its dual code. Then we determine the regularity of the ideals of circuits and cocircuits of a signed graph, and prove an algebraic formula in terms of the multiplicity for the frustration index of an unbalanced signed graph. [ABSTRACT FROM AUTHOR]

We study the problem of computing a minimum cut in a simple, undirected graph and give a deterministic O(mlog2 n log log2 n) time algorithm. This improves on both the best previously known deterministic running time of O(mlog12 n) (Kawarabayashi and Thorup [J. ACM, 66 (2018), 4]) and the best previously known randomized running time of O(mlog3 n) (Karger [J. ACM, 47 (2000), pp. 46--76]) for this problem, though Karger's algorithm can be further applied to weighted graphs. Moreover, our result extends to balanced directed graphs, where the balance of a directed graph captures how close the graph is to being Eulerian. Our approach is using the Kawarabayashi and Thorup graph compression technique, which repeatedly finds low conductance cuts. To find these cuts they use a diffusion-based local algorithm. We use instead a flow-based local algorithm and suitably adjust their framework to work with our flow-based subroutine. Both flowand diffusion-based methods have a long history of being applied to finding low conductance cuts. Diffusion algorithms have several variants that are naturally local, while it is more complicated to make flow methods local. Some prior work has proven nice properties for local flow-based algorithms with respect to improving or cleaning up low conductance cuts. Our flow subroutine, however, is the first that both is local and produces low conductance cuts. Thus, it may be of independent interest. [ABSTRACT FROM AUTHOR]

The restricted edge-connectivity of a connected graph G , denoted by λ ′ (G) , if exists, is the minimum number of edges whose deletion disconnects the graph such that each connected component has at least two vertices. The Kronecker product of graphs G and H , denoted by G × H , is the graph with vertex set V (G × H) = V (G) × V (H) , where two vertices (u 1 , v 1) and (u 2 , v 2) are adjacent in G × H if and only if u 1 u 2 ∈ E (G) and v 1 v 2 ∈ E (H). In this paper, it is proved that λ ′ (G × K n) = m i n { (n − 1) ξ (G) + 2 (n − 2) , n (n − 1) λ ′ (G) } for any graph G and a complete graph K n with n ≥ 3 vertices, where ξ (G) is minimum edge-degree of G , and a sufficient condition such that G × K n is λ ′ -optimal is acquired. [ABSTRACT FROM AUTHOR]

Let H be a connected hypergraph with minimum degree δ (H) , edge connectivity λ (H) and vertex connectivity κ (H). An r -uniform hypergraph is p - partite if its vertex set can be partitioned into p parts, and every edge contains at most one vertex from each part, where p ≥ r ≥ 2. Volkmann (1989) and Topp and Volkmann (1993) established sufficient conditions on p -partite graphs G implying λ (G) = δ (G) or κ (G) = λ (G). In this paper we generalize these well-known results to p -partite uniform hypergraphs. [ABSTRACT FROM AUTHOR]

Jaeger in 1979 showed that every 4-edge-connected graph is supereulerian, graphs that have spanning eulerian subgraphs. Catlin in 1988 sharpened Jaeger's result by showing that every 4-edge-connected graph is collapsible, graphs that are contractible configurations of supereulerian graphs. To further study collapsible subgraphs of a 4-edge-connected graph, in Catlin et al. (2009), it is shown that every 4-edge-connected graph remains collapsible after removing any two edges. We prove the following. • [(i)] Every 4-edge-connected G contains two vertices x , y such that one of x and y has minimum degree in G and both G − x and G − y are collapsible. • [(ii)] Let G be a 4-edge-connected graph and let X ⊂ E (G) be an edge subset with | X | ≤ 3. Then G − X is collapsible if and only if X is not contained in a 4-edge-cut of G. • [(iii)] Let G be a 4-edge-connected graph and let X ⊂ E (G) be an edge subset with | X | ≤ 4. Then G − X is collapsible if and only if G − X is not contractible to a member in { K 2 c , K 2 , K 2 , 2 , K 2 , 3 , K 2 , 4 }. These extend former results of Jaeger (1979) and Catlin (1988). [ABSTRACT FROM AUTHOR]

Let H be a set of connected graphs. A graph is said to be H-free if it does not contain any member of H as an induced subgraph. In this paper, we characterize all pairs R, S such that every connected {R,S}-free graph has the same (vertex)-connectivity and edge-connectivity. [ABSTRACT FROM AUTHOR]

Abstract Let G be a graph and L (G) be its line graph. In 1969, Chartrand and Stewart proved that κ ′ (L (G)) ≥ 2 κ ′ (G) − 2 , where κ ′ (G) and κ ′ (L (G)) denote the edge connectivity of G and L (G) respectively. We show a similar relationship holds for the essential edge connectivity of G and L (G) , written κ e ′ (G) and κ e ′ (L (G)) , respectively. In this note, it is proved that if L (G) is not a complete graph and G does not have a vertex of degree two, then κ e ′ (L (G)) ≥ 2 κ e ′ (G) − 2. An immediate corollary is that κ (L 2 (G)) ≥ 2 κ (L (G)) − 2 for such graphs G , where the vertex connectivity of the line graph L (G) and the second iterated line graph L 2 (G) are written as κ (L (G)) and κ (L 2 (G)) respectively. [ABSTRACT FROM AUTHOR]

The arithmetic graph Vn is defined as a graph with its vertex set is the set con-sists of the divisors of n (excluding 1) where n is a positive integer and n = pa11 pa22 pa33 . . . parr where p′is are distinct primes and ai's ≥ 1 and two distinct vertices a, b which are not of the same parity are adjacent in this graph if (a, b) = pi, for some i, 1 ≤ i ≤ r . In this paper, we study some results related to the connectivity K of an arithmetic graph. It is also shown that, the edge connectivity K′ and the connectivity K are equal in arithmetic graph Vn. [ABSTRACT FROM AUTHOR]

Let G be a graph of order n. The incidence energy IE(G) of graph G, IE for short, is defined as the sum of all singular values of its incidence matrix. In this paper, we determine the maximal incidence energy IE(G) among all connected graphs with the connectivity κ and edge connectivity κ' and Kκ' ∨(K1∪Kn-κ'-1) has the maximal incidence energy. [ABSTRACT FROM AUTHOR]

In recent years, the vertex connectivity and edge connectivity have aroused great interest of many scholars in the field of mathematics. We consider in this note that edge connectivity of a graph called k ′(G). In this note, If G is a bipartite graph with vertices n ≥ 3 we prove that k ′(G × Kn) = min {n(n - 1) k ′(G), (n - 1) δ(G)}, and their edge connectivity is satisfied k ′(G) = δ(G), we can obtain k ′(G × Kn) = δ (G × Kn) = (n - 1) δ(G). [ABSTRACT FROM AUTHOR]

The generalized A B C index of a graph G , denoted by A B C α ( G ) , is defined as the sum of weights ( d i + d j − 2 d i d j ) α over all edges v i v j of G , where α is an arbitrary non-zero real number, and d i is the degree of vertex v i of G . In this paper, we first prove that the generalized A B C index of a connected graph will increase with addition of edge(s) if α < 0 or 0 < α ≤ 1 ∕ 2 , which provides a useful tool for the study of extremal properties of the generalized A B C index. By means of this result, we then characterize the graphs having the maximal A B C α value for α < 0 among all connected graphs with given order and vertex connectivity, edge connectivity, or matching number. Our work extends some previously known results. [ABSTRACT FROM AUTHOR]

For an undirected simple graph G , the line graph L ( G ) is the graph whose vertex set is in one-to-one correspondence with the edge set of G where two vertices are adjacent if their corresponding edges in G have a common vertex. The energy E ( G ) is the sum of the absolute values of the eigenvalues of G . The vertex connectivity κ ( G ) is referred as the minimum number of vertices whose deletion disconnects G . The positive inertia ν + ( G ) corresponds to the number of positive eigenvalues of G . Finally, the matching number β ( G ) is the maximum number of independent edges of G . In this paper, we derive a sharp upper bound for the energy of the line graph of a graph G on n vertices having a vertex connectivity less than or equal to k . In addition, we obtain upper bounds on E ( G ) in terms of the edge connectivity, the inertia and the matching number of G . Moreover, a new family of hyperenergetic graphs is obtained. [ABSTRACT FROM AUTHOR]

Denote the sum of squares of all distances between all pairs of vertices in G by S ( G ) . In this paper, sharp bounds on the S ( G ) are determined for several classes of connected bipartite graphs. All the extremal graphs having the minimal S ( G ) in the class of all connected n -vertex bipartite graphs with a given vertex connectivity (resp. edge-connectivity) can be identified. [ABSTRACT FROM AUTHOR]