GIMS: Image matching system based on adaptive graph construction and graph neural network.

Hitting times of random walks on unicyclic graphs with a given number of pendant vertices

Asymptotic bounds on the numbers of vertices of polytopes of polystochastic matrices

Vertex-degree-based topological indices of uniform directed hypergraphs

On a lower bound for the number of vertices of an integral graph with a given diameter

Let G be a graph with n vertices and m edges, and let ρ1,ρ2, . . . ,ρn be the eigenvalues of the Randić matrix. The Randić Estrada index of G is REE(G) = Σni =1 eρi . In this paper, we establish bounds for the Randić Estrada index in terms of graph invariants such as the number of vertices and some Randić eigenvalues of graphs and improve some previously published lower bounds.

Let G be a graph with n vertices and m edges, and let ρ1,ρ2, . . . ,ρn be the eigenvalues of the Randić matrix. The Randić Estrada index of G is REE(G) = Σni =1 eρi. In this paper, we establish bounds for the Randić Estrada index in terms of graph invariants such as the number of vertices and some Randić eigenvalues of graphs and improve some previously published lower bounds.

The input of the popular roommates problem consists of a graph G = (V, E) and for each vertex v in V, strict preferences over the neighbors of v. Matching M is more popular than M' if the number of vertices preferring M to M' is larger than the number of vertices preferring M' to M. A matching M is called popular if there is no matching M' that is more popular than M. Faenza et al. (2019) and Gupta et al. (2021) proved that determining the existence of a popular matching in a popular roommates instance is NP-complete. In this paper we identify a class of instances that admit a polynomial-time algorithm for the problem. We also test these theoretical findings on randomly generated instances to determine the existence probability of a popular matching in them.

Let G be a connected simple graph. A dominating set S⊆ V(G) is a fair dominating set in G if S=V(G) or if S≠V(G) and all vertices not in S are dominated by the same number of vertices from S, that is, |N(u)∩ S|=|N(v)∩ S|>0 for every two vertices u,v∈ V(G)∖S. A fair dominating set S of V(G) is a secure fair dominating set of G if for each u∈V(G)∖S, there exists v∈S such that uv∈E(G) and the set (S∖{v})∪ {u} is a fair dominating set of G. The minimum cardinality of a secure fair dominating set of G, denoted by γ_sfd (G), is called the secure fair domination number of G. In this paper, we initiate a study of secure fair domination in graphs and give some important results.

Graph energy is one of the most significant aspects in mathematical chemistry that has generated a lot of interest in researchers over a long period of time. The concept of energy of a vertex, introduced by Arizmendi et al., provides a local spectral perspective to the graph energy, highlighting the significance of individual vertex energies and their contribution to the total energy of the graph, and is therefore becomes important to understand the role of each vertex in understanding the spectral properties of the graph. In literature, various authors have worked on studying the properties of vertex energy and computing the same for various classes of graphs using the method explained in the original article. Particularly, computing the vertex energies of integral graphs is commonly found in recent studies as it involves graphs with only integer eigenvalues, thereby ensuring efficiently obtaining exact values of vertex energies and not numerical approximations. Recently, Gutman et al. have proposed a novel approach to compute the vertex energy of a graph using eigenvalues and eigenvectors of a graph with a specified number of vertices. In the present study, we adopt this innovative approach, particularly useful with graphs having too many distinct eigenvalues and large number of vertex symmetries, to determine the vertex energies for all 22 connected integral graphs of order eight by finding all the eigenvalues and their corresponding orthonormal eigenvectors using the Gram-Schmidt process.

A proper t-edge-coloring of a graph G is a mapping $\alpha: E(G)\rightarrow \{1,\ldots,t\}$ such that all colors are used, and $\alpha(e)\neq \alpha(e^{\prime})$ for every pair of adjacent edges $e,e^{\prime}\in E(G)$. If $\alpha $ is a proper edge-coloring of a graph G and $v\in V(G)$, then the spectrum of a vertex v, denoted by $S\left(v,\alpha \right)$, is the set of all colors appearing on edges incident to v. The deficiency of $\alpha$ at vertex $v\in V(G)$, denoted by $\mathrm{def}(v,\alpha)$, is the minimum number of integers that must be added to $S\left(v,\alpha \right)$ to form an interval, and the deficiency $\mathrm{def}\left(G,\alpha\right)$ of a proper edge-coloring $\alpha$ of G is defined as the sum $\displaystyle\sum_{v\in V(G)}\mathrm{def}(v,\alpha)$. The deficiency of a graph G, denoted by $\mathrm{def}(G)$, is defined as follows: $\mathrm{def}(G)=\min_{\alpha}\mathrm{def}\left(G,\alpha\right)$, where the minimum is taken over all possible proper edge-colorings of G. In 2019, Davtyan, Minasyan, and Petrosyan provided an upper bound on the deficiency of complete multipartite graphs. In this paper, we improve this bound for complete tripartite and some complete 4-partite graphs. We also confirm the conjecture that states the deficiency of a graph is bounded by the number of vertices of the graph for all tripartite graphs containing up to 10 vertices.

Using spectral techniques, H. Huang proved that every subgraph of H n , the hypercube of dimension n , induced on more than half the vertices has maximum degree at least \(\sqrt {n} \) . Combined with earlier work, this completed a proof of the sensitivity conjecture. In this work we show how to derive Huang’s result using linear dependency and independence of vectors associated with the vertices of the hypercube. Our approach leads to several improvements of Huang’s result. In particular, we prove that in any induced subgraph of H n with more than half the number of vertices, there are two vertices, one of odd parity and the other of even parity, each with at least n vertices at distance at most 2. As an application, we show that for any Boolean function f , the polynomial degree of f is bounded above by \(\rm {s}_0(f) \rm {s}_1(f) \) , a statement which implies the sensitivity conjecture (but not immediately implied by the sensitivity conjecture). Using these linear dependencies, we show structural relations about the neighborhoods on the induced subgraphs at distance at most three. A key implement in Huang’s proof is to assign signs (+, −) to the edges of H n such that the product of the signs on each 4-cycle is −. With the set of negative edges being called a signature, one may observe that there are a total of \(2^{2^n-1} \) such signatures on H n satisfying this condition and that the symmetric difference of any two such signatures is an edge cut. A question of high interest then is to find the smallest size among all these signatures. This is known as the frustration index in the study of signed graphs. Here we provide lower and upper bounds for this parameter, observing that the two bounds match when n is a power of 4. We then establish a strong connection with other studies: On the one hand with a question of Erdős on the number of edges of a largest 4-cycle free subgraph of the hypercube. On the other hand with Ambainis functions which are used to show a separation between degree and adversary lower bounds on query complexity.

We investigate a number-theoretic graph labeling known as k-prime cordial labeling, where each vertex of a graph G is assigned a label from the set {1,2, ... , k}, and each edge receives a label equal to the greatest common divisor of its endpoint labels. A weak labeling is k-prime cordial if the number of vertices labeled with each integer differs by at most two, and the number of edges labeled 1 differs from those not labeled 1 by at most two. In this paper, we introduce the concept of the $\ell$-fold of a graph, denoted T_l(G), constructed by joining corresponding vertices across l copies of a base graph G. We focus on the case where G is a cycle graph C_n and show that T_l(C_n) admits a 4-prime cordial weak labeling for all l≥2. This result extends previous work on trigraphs and contributes to the broader understanding of cordial labeling in replicated graph structures.

In graph theory, understanding the labeling of graphs and hypergraphs provides valuable insights into their structural properties and applications. A hypergraph generalizes the notion of a conventional graph, defined as a mathematical structure built from a vertex set V and a hyperedge set E, where each hyperedge is allowed to connect two or more vertices simultaneously. The essential distinction between a graph and a hypergraph lies in their edges. While in a graph a single edge connects exactly two vertices, in a hypergraph a single hyperedge may connect any number of vertices, including two. A hypergraph is considered to admit a super (a, d) -hyperedge antimagic total labeling, such that the vertex label functions f: V(H)  1, 2, 3, ....., V(H) then f: E(H)  V(H) + 1, ....., V(H) + V(H) and weight w(ei) = ∑ f(ei) + ∑ f(Vi,j), where i denotes the number of hyperedges, j represents the number of vertices contained in a hyperedge, and e_i refers to the set of vertices and its associated edges with weight w(ei) for each hyperedge. A super (a, d) -hyperedge antimagic total labeling is formulated as a labeling scheme based on arithmetic progressions, where ???? serves as the initial value and d denotes the common difference between consecutive labels. In this scheme, the total weight of a hyperedge is determined by deriving from the sum of the vertex labels and the label of the respective hyperedge. The labels are arranged in an arithmetic sequence, ensuring that each hyperedge has a distinct weight. This study focuses on several special classes of hypergraphs, namely, the volcano graph, the semi-parachute graph, and the comb product of graphs, to implement and examine the characteristics of the super (a, d)-hyperedge antimagic total labeling. By focusing on these graph classes, the study contributes to combinatorics by offering a deeper understanding of hypergraph labeling schemes and their potential applications in network theory, coding theory, and data modeling.

Let G be a simple, connected, undirected, nontrivial graph and let c be a coloring of the vertices of G using the colors red and white such that not all the vertices are colored white. For each vertex v of G , we can assign a vector code d⃗ (v)=(a1,a2,…,adiam(G)) , where diam(G) is the diameter of G and, for each i=1,2,…,diam(G) , the component ai is equal to the number of red vertices whose distance to v is i . Then, c is said to be an ID-coloring of G if d⃗ (v)≠d⃗ (w) for any pair of distinct vertices v and w . If G has an identification coloring, it is called an ID-graph and its identification number ID(G) is defined to be the least number of red vertices needed to construct an ID-coloring of G . An ID-coloring of G can be used to distinguish its vertices from each other and this is particularly interesting when G has nontrivial automorphisms. Indeed, identification colorings have been studied for paths, cycles, grids, prisms, and some tree families, including caterpillar and lobster graphs. In this paper, we consider identification colorings for antiprism graphs, which have not been previously studied in relation to this topic. An antiprism graph is a graph that corresponds to the skeleton of an antiprism. Antiprism graphs are known to be vertex-transitive; thus, they possess nontrivial automorphisms. Our results include a characterization of all ID-antiprisms and the determination of the identification number of any ID-antiprism.

The centrality of a node is often used to measure its importance to the structure of a network. Some centrality measures can be extended to measure the importance of a path. In this paper, we consider the problem of finding the most central shortest path. We show that the computational complexity of this problem depends on the measure of centrality used and in the case of degree centrality, whether the network is weighted or not. We develop a polynomial algorithm for the most degree-central shortest path problem with the worst-case running time of [Formula: see text], where | V | is the number of vertices, | E | is the number of edges, and [Formula: see text] is the maximum degree of the graph. In addition, we show that the same problem is NP-hard on a weighted graph. Furthermore, we show that the problem of finding the most betweenness-central shortest path is solvable in polynomial time, whereas finding the most closeness-central shortest path is NP-hard, regardless of whether the graph is weighted or not. We also develop an algorithm for finding the most betweenness-central shortest path with a running time of [Formula: see text] on unweighted graphs and [Formula: see text] on graphs with positively weighted edges. To conclude our paper, we conduct a numerical study of our algorithms on synthetic and real-world networks and compare our results with the existing literature. History: Accepted by Russell Bent, Area Editor for Network Optimization: Algorithms and Applications. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0945 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0945 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Abstract Let G G be a simple graph on n n vertices with degree sequence d 1 , … , d n {d}_{1},\ldots ,{d}_{n} . Fajtlowicz ( On conjectures of Graffiti , Discrete Math. 72 (1988), 113–118) defined the temperature of a vertex v v of G G as d n − d \frac{d}{n-d} , where d d is the degree of v v . Motivated by this definition, we define the temperature index of G G , denoted by T ( G ) T\left(G) , as T ( G ) = d 1 n − d 1 + ⋯ + d n n − d n T\left(G)=\frac{{d}_{1}}{n-{d}_{1}}+\cdots +\frac{{d}_{n}}{n-{d}_{n}} . We obtain some lower bounds and upper bounds for T ( G ) T\left(G) in terms of the number of vertices, the number of edges, the maximum and the minimum vertex degree and the Zagreb index ( Z ( G ) = d 1 2 + ⋯ + d n 2 Z\left(G)={d}_{1}^{2}+\cdots +{d}_{n}^{2} ). Using these results we derive new bounds for the Zagreb index of graphs. Finally, we study the temperature index of graphs from the point of view of spectra of graphs (the eigenvalues of their adjacency matrices). In particular, we show that G G has at least one eigenvalue in the interval [ − n − δ T ( G ) − 2 m n , n − δ T ( G ) − 2 m n ] \left[-\sqrt{n-\delta }\sqrt{T\left(G)-\frac{2m}{n}},\sqrt{n-\delta }\sqrt{T\left(G)-\frac{2m}{n}}] .

Network data are commonly collected in a variety of applications, representing either directly measured or statistically inferred connections between subjects or features of interest. In an increasing number of domains, these networks are collected over time, such as repeated interactions between users of a social media platform, or across multiple subjects, such as in multi-subject neuroimaging studies. When analyzing multiple large networks, dimensionality reduction techniques are often used to embed networks in a more tractable low-dimensional space. To this end, we develop a framework for principal components analysis (PCA) on collections of networks via a specialized tensor decomposition, termed Semi-Symmetric Tensor PCA or SST-PCA, and analyze it theoretically. Notably, we show that SST-PCA achieves the same accuracy as classical matrix PCA, with error proportional to the square root of the number of vertices and not the number of edges as might be expected. Our framework inherits many of the strengths of classical PCA and is suitable for a wide range of unsupervised learning tasks, including identifying principal networks, isolating changepoints and outliers, and for characterizing the “variability network” of the most varying edges. Finally, we demonstrate the effectiveness of SST-PCA in simulation and on an example from empirical social studies.

The Ising model is important in statistical modeling and inference in many applications, however, its normalizing constant, mean number of active vertices and mean spin interaction—quantities often needed in inference—are computationally intractable. We provide accurate approximations that make it possible to numerically calculate these quantities in the homogeneous case. Simulation studies indicate good performance of our approximation formulae that are scalable and unfazed by the size (number of nodes, degree of graph) of the Markov Random Field. The practical import of our approximation formulae is illustrated in performing Bayesian inference in a functional Magnetic Resonance Imaging activation detection experiment, in likelihood ratio testing, for anisotropy in the spatial patterns of yearly increases in pistachio tree yields, and for independence of the least significant bit in the three color channels of a gigapixel image. Supplementary materials for this article are available online.

Leveraging the distinct properties of X-rays and γ-rays, a novel radiation therapy platform integrating both modalities has been implemented in clinical practice (NMPA: 20223050973; FDA: K210921). This study investigates the application of this integrated approach in spatially fractionated radiotherapy, systematically evaluating its feasibility and therapeutic potential. In this retrospective study, lattice radiotherapy (LRT) was designed for 10 NSCLC cases with gross tumor volumes (GTV) ranging from 572 to 1367cm³ (mean 862.9 ± 285.4cm³), incorporating the number of high-dose vertices per case ranged from 6 to 13, with a median of 8.5, respectively. Each LRT plan consisted of a single 12Gy dose to the intratumoral vertices, followed by conventional external beam radiotherapy (cEBRT) delivering 25 daily fractions of 1.8Gy to the Planning Target Volume (PTV). Treatment plans were developed using the Varian Eclipse 13.5 Treatment Planning System (TPS) for Linac plans, while TaiChiB system plans were generated using the RT PRO TPS: a focused gamma plan was created to target the vertices, and a Linac plan was optimized to cover the PTV. A comparative analysis of D0.5cc, D10/D90, EQD2, and Dmean was performed to evaluate the ability of dual-modality to optimize high-dose vertices while reducing doses to GTV margins and organs at risk (OARs). The LRT plan involved the placement of a median of 8.5 high-dose vertices (range, 6 to13), each with a diameter of 1.5cm and spaced 3-3.5cm apart within the GTV. The average vertices volume was 17.2 ± 4.5cm³, corresponding to 2.05% ± 0.34% of the GTV. Compared to the Linac plans, the TaiChiB system plans demonstrated significantly increased D0.5cc, Dmean, and EQD2 within the GTV (P < 0.01), improved peak/valley dose ratio (PVDR, D10/D90, P < 0.01), and reduced marginal GTV dose. Additionally, the TaiChiB system plans significantly reduced doses to OARs, including right lung Dmean (P = 0.031), heart Dmean (P = 0.024), esophagus Dmax (P < 0.01), and spinal cord Dmax (P = 0.042). All plans complied with the OARs dose constraints, thereby ensuring clinical feasibility and patient safety. By integrating X-ray and γ-ray technologies, this platform enhances the vertex dose within the GTV while reducing doses to the GTV margins and OARs, offering a promising and feasible approach for the treatment of LRT in patients with large-volume lung tumors.