Vertices Of Degree Research Articles

Random matrices with row constraints and eigenvalue distributions of graph Laplacians

Symmetric matrices with zero row sums occur in many theoretical settings and in real-life applications. When the offdiagonal elements of such matrices are i.i.d. random variables and the matrices are large, the eigenvalue distributions converge to a peculiar universal curve that looks like a cross between the Wigner semicircle and a Gaussian distribution. An analytic theory for this curve, originally due to Fyodorov, can be developed using supersymmetry-based techniques. We extend these derivations to the case of sparse matrices, including the important case of graph Laplacians for large random graphs with N vertices of mean degree c. In the regime , the eigenvalue distribution of the ordinary graph Laplacian (diffusion with a fixed transition rate per edge) tends to a shifted and scaled version of , centered at c with width . At smaller c, this curve receives corrections in powers of accurately captured by our theory. For the normalized graph Laplacian (diffusion with a fixed transition rate per vertex), the large c limit is a shifted and scaled Wigner semicircle, again with corrections captured by our analysis.

On the Hyper Zagreb Index of Trees with a Specified Degree of Vertices

Topological indices are the numerical descriptors that correspond to some certain physicochemical properties of a chemical compound such as the boiling point, acentric factor, enthalpy of vaporisation, heat of fusion, etc. Among these topological indices, the Hyper Zagreb index, is the most effectively used topological index to predict the acentric factor of some octane isomers. In the current work, we investigate the extremal values of the Hyper Zagreb index for some classes of trees.

Mathematical and chemical properties of geometry-based invariants and its applications

Recently, based on elementary geometry, Gutman proposed several geometry-based invariants (i.e., SO, SO1, SO2, SO3, SO4, SO5, SO6). The Sombor index was defined as SO(G)=∑uv∈E(G)du2+dv2, the first Sombor index was defined as SO1(G)=12∑uv∈E(G)|du2−dv2|, where du denotes the degree of vertex u.In this paper, we consider the mathematical properties of these geometry-based invariants. We determine the maximum trees (resp. unicyclic graphs) with given diameter, the maximum trees with given matching number, the maximum trees with given pendent vertices, the maximum trees (resp. minimum trees) with given branching number, the minimum trees with given maximum degree and second maximum degree, the minimum unicyclic graphs with given maximum degree and girth, the minimum connected graphs with given maximum degree and pendent vertices, and some properties of maximum connected graphs with given pendent vertices with respect to the first Sombor index SO1. As an application, we inaugurate these geometry-based invariants and verify their chemical applicability.

Subdomination in Graphs with Upper-Bounded Vertex Degree

We find a lower bound for the k-subdomination number on the set of graphs with a given upper bound for vertex degrees. We study the cases where the proposed lower bound is sharp, construct the optimal graphs and indicate the corresponding k-subdominating functions. The results are interpreted in terms of social structures.

On the Wiener index and the hyper-Wiener index of the Kragujevac trees

In this paper, the Wiener index and the hyper-Wiener index of the Kragujevac trees is computed in term of its vertex degrees. As application, we obtain an upper bond and a lower bound for the Wiener index and the hyper-Wiener index of these trees.

Evolution mechanism of the local network structure recorded in distribution of distances between neighbors of each vertex

The explosive development of research on real networks have led to the proposal of many mathematical network models explaining properties shared in the real networked systems. However, considering that different models can similarly reproduce the typical properties of networks, the estimation of the evolutionary process is more essential for appropriate network modeling than is the description of current network structures. In this paper, we propose a new measure “distribution of distances between neighbors (DDN) of each vertex” to investigate network development. We first examine several model networks, including growth models based on the attachment rule and on the local rule and their hybrid models, and show that the DDN profiles of these networks accurately reflect the differences in growth mechanisms. In particular, for the local rule model, the dependence of the mean DDN of vertices on the vertex degree records the evolution of the local network structure. The application of the same analysis to several real networks show that they share the characteristics of DDN profiles with the local rule and hybrid models. These results imply that these model networks appropriately describe the evolutionary processes of real networks. In addition, we examine the characteristics of DDN specific to fractal and non-fractal networks, and show that the presence or absence of fractality of some real networks can be explained by the influence of long-range edges on the local rule. An exception to this is the co-authorships network, which is expected to be generated by connecting several fractal sub-networks.

Gallai's Path Decomposition for 2-degenerate Graphs

Gallai's path decomposition conjecture states that if $G$ is a connected graph on $n$ vertices, then the edges of $G$ can be decomposed into at most $\lceil \frac{n }{2} \rceil$ paths. A graph is said to be an odd semi-clique if it can be obtained from a clique on $2k+1$ vertices by deleting at most $k-1$ edges. Bonamy and Perrett asked if the edges of every connected graph $G$ on $n$ vertices can be decomposed into at most $\lfloor \frac{n}{2} \rfloor$ paths unless $G$ is an odd semi-clique. A graph $G$ is said to be 2-degenerate if every subgraph of $G$ has a vertex of degree at most $2$. In this paper, we prove that the edges of any connected 2-degenerate graph $G$ on $n$ vertices can be decomposed into at most $\lfloor \frac{n }{2} \rfloor$ paths unless $G$ is a triangle.

Connectivity index of directed rough fuzzy graphs and its application in traffic flow network

The directed rough fuzzy graph (DRFG) is a fusion of rough and fuzzy theory, as it deals with incomplete and vague information simultaneously. Connection or the strength of connectivity $$(\mathcal{S}\mathcal{C})$$ is vital in the realm of circuits or networks that are linked to the real world. As a result, $$\mathcal{S}\mathcal{C} $$ is one of the most essential aspects of a directed rough fuzzy network system. The neighborhood connectivity index $$(\mathcal {NCI})$$ is one such parameter that has a variety of applications in network theory. In this study, our main objective is to present a new topological index $$\mathcal {NCI}$$ based on DRFGs to solve complicated problems. Motivated by the modeling of networks, the strength of vertices to their neighboring vertices, and the efficiency of DRFGs to solve complex problems, we aim to study the NCI of DRFGs. In this paper, we successfully introduce a notion NCI based on DRFGs to deal with the uncertainties that arise in real-world problems. Based on the strength of vertices to neighboring vertices, we provide several lower and upper bounds for the $$\mathcal {NCI}$$ of DRFGs with reference to other graph invariants such as the number of vertices, edges, and degree distance. When we study $$\mathcal {NCI}$$ in operations for DRFGs with a large number of vertices, the degree of vertices in a DRFG provides a confusing picture. Therefore, a mechanism to determine the $$\mathcal {NCI}$$ for DRFG operations is therefore required. Therefore, generalized formulas for the $$\mathcal {NCI}$$ of DRFGs obtained by operations such as union, composition, and Cartesian product are also developed. An algorithm for obtaining the $$\mathcal {NCI}$$ of DRFGs is also proposed. Further, an application of the $$\mathcal {NCI}$$ of DRFGs in traffic flow networks was discussed to identify the busiest intersection using the proposed algorithm. Finally, we illustrate a comparative analysis and analysis table of the established approach ( $$\mathcal {NCI}$$ ) with existing techniques (connectivity index $$(\mathcal{C}\mathcal{I})$$ and Wiener index $$(\mathcal{W}\mathcal{I})$$ ) are shown to demonstrate the validity of the presented approach.

Unimodality and monotonic portions of certain domination polynomials

Given a graph G on n vertices, a subset of vertices U⊆V(G) is dominating if every vertex of V(G) is either in U or adjacent to a vertex of U. The domination polynomial of G is the generating function whose coefficients are the number of dominating sets of a given size. In 2014, Alikhani and Peng conjectured that the domination polynomial is unimodal, i.e., its coefficients are non-decreasing and then non-increasing. We prove unimodality for spider graphs with at most 400 legs (of arbitrary length), lollipop graphs, arbitrary direct products of complete graphs, and Cartesian products of two complete graphs. We show that for every graph, a portion of the coefficients are non-increasing, and, for certain graphs with low upper domination number, this is sufficient to prove unimodality. Furthermore, we show that for graphs with m universal vertices, i.e., vertices of degree n−1, the last 2−m−1(2m−1)n coefficients of their domination polynomial are non-increasing.

A (Slightly) Improved Approximation Algorithm for Metric TSP

In “An Improved Approximation Algorithm for TSP,” Karlin, Klein, and Oveis Gharan design the first improvement over the classical 1.5 approximation algorithm of Christofides-Serdyukov after more than 40 years. Their algorithm first chooses a random spanning tree from the maximum entropy distribution of spanning trees with marginals equal to the optimum LP solution of TSP, and then, similar to Christofides’ algorithm, it adds the minimum cost matching on the odd degree vertices of the tree. To analyze their simple algorithms, they prove and exploit new tools from the theory of strongly Rayleigh distributions.

Contagion risks and security investment in directed networks.

We develop a model for contagion risks and optimal security investment in a directed network of interconnected agents with heterogeneous degrees, loss functions, and security profiles. Our model generalizes several contagion models in the literature, particularly the independent cascade model and the linear threshold model. We state various limit theorems on the final size of infected agents in the case of random networks with given vertex degrees for finite and infinite-variance degree distributions. The results allow us to derive a resilience condition for the network in response to the infection of a large group of agents and quantify how contagion amplifies small shocks to the network. We show that when the degree distribution has infinite variance and highly correlated in- and out-degrees, even when agents have high thresholds, a sub-linear fraction of initially infected agents is enough to trigger the infection of a positive fraction of nodes. We also demonstrate how these results are sensitive to vertex and edge percolation (intervention). We then study the asymptotic Nash equilibrium and socially optimal security investment. In the asymptotic limit, agents' risk depends on all other agents' investments through an aggregate quantity that we call network vulnerability. The limit theorems enable us to capture the impact of one class of agents' decisions on the overall network vulnerability. Based on our results, the vulnerability is semi-analytic, allowing for a tractable Nash equilibrium. We provide sufficient conditions for investment in equilibrium to be monotone in network vulnerability. When investment is monotone, we demonstrate that the (asymptotic) Nash equilibrium is unique. In the specific example of two types of core-periphery agents, we illustrate the strong effect of cost heterogeneity on network vulnerability and the non-monotonous investment as a function of costs.

On the inverse Collatz-Sinogowitz irregularity problem

Abstract The Collatz-Sinogowitz irregularity index is the oldest known numerical measure of graph irregularity. For a simple and connected graph G G of order n n and size m m , it is defined as CS ( G ) = λ 1 − 2 m / n , \hspace{0.1em}\text{CS}\hspace{0.1em}\left(G)={\lambda }_{1}-2m\hspace{0.1em}\text{/}\hspace{0.1em}n, where λ 1 {\lambda }_{1} is the largest eigenvalue of the adjacency matrix of G G , and 2 m / n 2m\hspace{0.1em}\text{/}\hspace{0.1em}n is the average vertex degree of G G . Here, the Collatz-Sinogowitz inverse irregularity problem is studied. For every integer i ≥ 0 i\ge 0 , it is shown that there exists a graph G G such that CS ( G ) = i \hspace{0.1em}\text{CS}\hspace{0.1em}\left(G)=i . Also, for every interval I i = ( i , i + 1 ) {I}_{i}=\left(i,i+1) , it is shown that there are infinitely many graphs whose Collatz-Sinogowitz irregularity lies in I i {I}_{i} .

On bounds of Aα-eigenvalue multiplicity and the rank of a complex unit gain graph

Let Φ=(G,φ) be a connected complex unit gain graph (T-gain graph) of n vertices with largest vertex degree Δ, adjacency matrix A(Φ), and degree matrix D(Φ). Let mα(Φ,λ) be the multiplicity of λ as an eigenvalue of Aα(Φ):=αD(Φ)+(1−α)A(Φ), for α∈[0,1). In this article, we establish that mα(Φ,λ)≤(Δ−2)n+2Δ−1 and characterize the sharpness. Then, we obtain some lower bounds for the rank r(Φ) in terms of n and Δ including r(Φ)≥n−2Δ−1 and characterize their sharpness. Besides, we introduce zero-2-walk gain graphs and study their properties. It is shown that a zero-2-walk gain graph is always regular. Furthermore, we prove that Φ has exactly two distinct eigenvalues with equal magnitude if and only if it is a zero-2-walk gain graph. Using this, we establish a lower bound of r(Φ) in terms of the number of edges and characterize the sharpness. Result about mα(Φ,λ) extends the corresponding known result for undirected graphs and simplifies the existing proof, and other bounds of r(Φ) obtained in this article work better than the bounds given elsewhere.

Enumeration of Multi-rooted Plane Trees

We give closed form expressions for the numbers of multi-rooted plane trees with specified degrees of root vertices. This results in an infinite number of integer sequences, some of which are known to have an alternative interpretation. We also propose recursion relations for numbers of such trees as well as for the corresponding generating functions. Explicit expressions for the generating functions corresponding to plane trees having two and three roots are derived. As a by-product, we obtain a new binomial identity and a conjecture relating hypergeometric functions.

Sharp Bounds on the Generalized Multiplicative First Zagreb Index of Graphs with Application to QSPR Modeling

Degree sequence measurements on graphs have attracted a lot of research interest in recent decades. Multiplying the degrees of adjacent vertices in graph Ω provides the multiplicative first Zagreb index of a graph. In the context of graph theory, the generalized multiplicative first Zagreb index of a graph Ω is defined as the product of the sum of the αth powers of the vertex degrees of Ω, where α is a real number such that α≠0 and α≠1. The focus of this work is on the extremal graphs for several classes of graphs including trees, unicyclic, and bicyclic graphs, with respect to the generalized multiplicative first Zagreb index. In the initial step, we identify a set of operations that either increases or decreases the generalized multiplicative first Zagreb index for graphs. We then involve analysis of the generalized multiplicative first Zagreb index achieving sharp bounds by characterizing the maximum or minimum graphs for those classes. We present applications of the generalized multiplicative first Zagreb index Π1α for predicting the π-electronic energy Eπ(β) of benzenoid hydrocarbons. In particular, we answer the question concerning the value of α for which the predictive potential of Π1α with Eπ for lower benzenoid hydrocarbons is the strongest. In fact, our statistical analysis delivers that Π1α correlates with Eπ of lower benzenoid hydrocarbons with correlation coefficient ρ=−0.998, if α=−0.00496. In QSPR modeling, the value ρ=−0.998 is considered to be considerably significant.

Graphs of fixed order and size with maximal Aα-index

For any real number α∈[0,1], by the Aα-matrix of a graph G we mean the matrix Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and the diagonal matrix of vertex degrees of G, respectively. The largest eigenvalue of Aα(G) is called the Aα-index of G. In this paper, we settle the problem of characterizing graphs which attain the maximum Aα-index over G(n,n+k), the class of graphs with n vertices and n+k edges, for −1≤k≤n−3 and 12≤α<1. The following result is obtained: for −1≤k≤n−3, when 12≤α<1, Hn,k is the unique graph in G(n,n+k) that maximizes the Aα-index, except when (n,k)=(4,−1),(n,2) or (7,3) and α=12, or (n,k)=(5,1) and α∈[12,35−40924]. When (n,k,α)=(4,−1,12), the optimal graphs are H4,−1 and K3∪K1; when (n,k,α)=(n,2,12), the optimal graphs are Hn,2 and Gn,2; when (n,k,α)=(5,1,35−40924), the optimal graphs are H5,1 and K4∪K1; when (n,k,α)=(7,3,12), the optimal graphs are H7,3 and K5∪2K1; when (n,k)=(5,1) and 12≤α<35−40924, K4∪K1 is the unique graph that maximizes the Aα-index. Our work completes the corresponding work of Chang and Tam (2010) and Zhai et al. (2022) for the special case α=12. As a by-product, we provide a new proof for the known result that for any positive integer m and any real number α∈[12,1), if (m,α)≠(3,12), then a graph maximizes the Aα-index over all graphs with m edges if and only if it is the union of K1,m with a (possibly empty) null graph; a graph maximizes the A12-index over all graphs with three edges if and only if it is the union of K1,3 or K3 with a (possibly empty) null graph. Some open questions are also posed.

Swift chiral quantum walks

A continuous-time quantum walk (CTQW) is sedentary if the return probability in the starting vertex is close to one at all times. Recent results imply that, when starting from a maximal degree vertex, the CTQW dynamics generated by the Laplacian and adjacency matrices are typically sedentary. In this paper, we show that the addition of appropriate complex phases to the edges of the graph, defining a chiral CTQW, can cure sedentarity and lead to swift chiral quantum walks of the adjacency type, which bring the return probability to zero in the shortest time possible. We also provide a no-go theorem for swift chiral CTQWs of the Laplacian type. Our results provide one of the first, general characterization of tasks that can and cannot be achieved with chiral CTQWs.

Improved bound on vertex degree version of Erdős matching conjecture

AbstractFor a ‐uniform hypergraph , let denote the minimum vertex degree of , and denote the size of the largest matching in . In this paper, we show that for any and , there exists an integer such that for positive integers and , if is an ‐vertex ‐graph with , then . This improves upon earlier results of Bollobás, Daykin, and Erdős for the range and Huang and Zhao for the range .

On the Spectral Radius of Minimally 2-(Edge)-Connected Graphs with Given Size

A graph is minimally $k$-connected ($k$-edge-connected) if it is $k$-connected ($k$-edge-connected) and deleting any arbitrary chosen edge always leaves a graph which is not $k$-connected ($k$-edge-connected). Let $m= \binom{d}{2}+t$, $1\leq t\leq d$ and $G_m$ be the graph obtained from the complete graph $K_d$ by adding one new vertex of degree $t$. Let $H_m$ be the graph obtained from $K_d\backslash\{e\}$ by adding one new vertex adjacent to precisely two vertices of degree $d-1$ in $K_d\backslash\{e\}$. Rowlinson [Linear Algebra Appl., 110 (1988) 43--53.] showed that $G_m$ attains the maximum spectral radius among all graphs of size $m$. This classic result indicates that $G_m$ attains the maximum spectral radius among all $2$-(edge)-connected graphs of size $m=\binom{d}{2}+t$ except $t=1$. The next year, Rowlinson [Europ. J. Combin., 10 (1989) 489--497] proved that $H_m$ attains the maximum spectral radius among all $2$-connected graphs of size $m=\binom{d}{2}+1$ ($d\geq 5$), this also indicates $H_m$ is the unique extremal graph among all $2$-connected graphs of size $m=\binom{d}{2}+1$ ($d\geq 5$). Observe that neither $G_m$ nor $H_m$ are minimally $2$-(edge)-connected graphs. In this paper, we determine the maximum spectral radius for the minimally $2$-connected ($2$-edge-connected) graphs of given size; moreover, the corresponding extremal graphs are also characterized.

Sudoku number of graphs

We introduce a concept in graph coloring motivated by the popular Sudoku puzzle. Let be a graph of order n with chromatic number and let Let be a k-coloring of the induced subgraph The coloring is called an extendable coloring if can be extended to a k-coloring of G. We say that is a Sudoku coloring of G if can be uniquely extended to a k-coloring of G. The smallest order of such an induced subgraph of G which admits a Sudoku coloring is called the Sudoku number of G and is denoted by In this paper we initiate a study of this parameter. We first show that this parameter is related to list coloring of graphs. In Section 2, basic properties of Sudoku coloring that are related to color dominating vertices, chromatic numbers and degree of vertices, are given. Particularly, we obtained necessary conditions for being extendable, and for being a Sudoku coloring. In Section 3, we determined the Sudoku number of various families of graphs. Particularly, we showed that a connected graph G has sn(G) = 1 if and only if G is bipartite. Consequently, every tree T has sn(T) = 1. We also proved that if and only if G = Kn . Moreover, a graph G with small chromatic number may have arbitrarily large Sudoku number. In Section 4, we proved that extendable partial coloring problem is NP-complete. Extendable coloring and Sudoku coloring are nice tools for providing a k-coloring of G.