Narrow Results

Let $G$ be a finite group. The identity graph $G$, denoted by ${\mathrm{\Gamma }}_G,$ is a simple, undirected graph with vertex set equal to $G$ and two distinct vertices $g$ and $h$ are said to be adjacent if and only if $gh=g$ or $h$ or $e$. This paper discusses the spectra of adjacency, Laplacian, signless Laplacian, distance, distance Laplacian, and distance signless Laplacian matrices of ${\mathrm{\Gamma }}_G$.

In this paper, we offer a simple combinatorial approach for calculating the distance matrix, distance-based structural polynomials and many distance-based indices of the zero-divisor graph $\mathrm{\Gamma }(R)$ associated to a finite reduced ring $R={𝔽}_q\times {𝔽}_q\times \cdots \times {𝔽}_q,$ where ${𝔽}_q$ is a finite field. A key contribution is the development of an efficient algorithm that computes the exponentially large-sized distance matrix of $\mathrm{\Gamma }(R)$ in terms of smaller-sized variants of the distance matrix. By utilizing the distance matrix, we derive explicit formulas for important distance-based graph invariants of $\mathrm{\Gamma }(R)$ such as the Hosoya polynomial, Harary polynomial, and Harary index, Wiener index and its variants. We achieve this by formulating recurrence relations using the distance matrix and solving them combinatorially. Consequently, for the Boolean graph and the zero-divisor graph $\mathrm{\Gamma }({\mathbb{Z}}_p\times {\mathbb{Z}}_p\times \cdots \times {\mathbb{Z}}_p)$ ($p$ a prime), the distance matrix and the distance-based invariants above can be explicitly obtained. Moreover, we establish an upper bound for the Wiener index of $\mathrm{\Gamma }({\mathbb{Z}}_m\times {\mathbb{Z}}_m\times \cdots \times {\mathbb{Z}}_m)$ and also show that this bound is attained if and only if $m$ is prime.

There are many community organizations in social and biological networks. How to identify these community structure in complex networks has become a hot issue. In this paper, an algorithm to detect community structure of networks is proposed by using spectra of distance modularity matrix. The proposed algorithm focuses on the distance of vertices within communities, rather than the most weakly connected vertex pairs or number of edges between communities. The experimental results show that our method achieves better effectiveness to identify community structure for a variety of real-world networks and computer generated networks with a little more time-consumption.

The genetic code is the rule by which DNA stores the genetic information about formation of protein molecule. In this paper, a partial ordering is equipped on the genetic code and a lattice structure has been developed from it. The codon–anticodon interaction, hydrogen bond number and the chemical types of bases play an important role in the partial ordering. We have established some relations between the lattice structure of the genetic code and physico-chemical properties of amino acids. Taking into consideration the evolutionary importance of base positions in codons we have constructed a distance matrix for the amino acids. Further with a real life example we have demonstrated the relationship between frequently occurring mutations and codon distances.

Nearly incompressible problems in three dimensions are the important problems in practical engineering computation. The volume-locking phenomenon will appear when the commonly used finite elements such as linear elements are applied to the solution of these problems. There are many efficient approaches to overcome this locking phenomenon, one of which is the higher-order conforming finite element method. However, we often use the lower-order nonconforming elements as Wilson elements by considering the computational complexity for three-dimensional (3D) problems considered. In general, the convergence of Wilson elements will heavily rely on the quality of the meshes. It will greatly deteriorate or no longer converge when the mesh distortion is very large. In this paper, the refined element method based on Wilson element is first applied to solve nearly incompressible elasticity problems, and the influence of mesh quality on the refined element is tested numerically. Its validity is verified by some numerical examples. By using the internal condensation method, the refined element discrete system of equations is deduced into the one which is spectrally equivalent to an 8-node hexahedral element discrete system of equations. And then, a type of efficient algebraic multigrid (AMG) preconditioner is presented by combining both the coarsening techniques based on the distance matrix and the effective smoothing operators. The resulting preconditioned conjugate gradient (PCG) method is efficient for 3D nearly incompressible problems. The numerical results verify the efficiency and robustness of the proposed method.

Although high quality multiple sequence alignment is an essential task in bioinformatics, it becomes a big dilemma nowadays due to the gigantic explosion in the amount of molecular data. The most consuming time and space phase is the distance matrix computation. This paper addresses this issue by proposing a vectorized parallel method that accomplishes the huge number of similarity comparisons faster in less space. Performance tests on real biological datasets using core-i7 show superior results in terms of time and space.

The distance, distance signless Laplacian and distance Laplacian matrix of a simple connected graph $G,$ are denoted by $D(G),D_Q(G)=D(G)+\mathrm{\text{Tr}}(G)$ and $D_L(G)=\mathrm{\text{Tr}}(G)-D(G),$ respectively, where $\mathrm{\text{Tr}}(G)$ is the diagonal matrix of vertex transmission. Ger$\check{\mathrm{\text{s}}}$gorin discs for any $n\times n$ square matrix $A=[a_{ij}]$ are the discs $\{z\in ℂ:|z-a_{ii}|\le R_i(A)\}$, where $R_i(A)={\sum }_{j\ne i}\left|a_{ij}\right|,i=1,2,\dots ,n$. The famous Ger$\check{\mathrm{\text{s}}}$gorin disc theorem says that all the eigenvalues of a square matrix lie in the union of the Ger$\check{\mathrm{\text{s}}}$gorin discs of that matrix. In this paper, some classes of graphs are studied for which the smallest Ger$\check{\mathrm{\text{s}}}$gorin disc contains every distance and distance signless Laplacian eigenvalues except the spectral radius of the corresponding matrix. For all connected graphs, a lower bound and for trees, an upper bound of every distance signless Laplacian eigenvalues except the spectral radius is given in this paper. These bounds are comparatively better than the existing bounds. By applying these bounds, we find some infinite classes of graphs for which the smallest Ger$\check{\mathrm{\text{s}}}$gorin disc contains every distance signless Laplacian eigenvalues except the spectral radius of the distance signless Laplacian matrix. For the distance Laplacian eigenvalues, we have given an upper bound and then find a condition for which the smallest Ger$\check{\mathrm{\text{s}}}$gorin disc contains every distance Laplacian eigenvalue of the distance Laplacian matrix. These results give partial answers from some questions that are raised in [2].

Let $Z(G)$ be the center of a finite non-abelian group $G.$ The non-commuting graph of $G$ is a simple undirected graph with vertex set $G\setminus Z(G),$ and two vertices $u$ and $v$ are adjacent if and only if $uv\ne vu.$ A group $G$ is called a CA-group if the centralizer $C_G(x)$ for every non central element $x$ is abelian. In this paper, we investigate the distance, distance (signless) Laplacian spectra of non-commuting graphs of some CA-groups, and obtain some conditions on them so that the corresponding non-commuting graph is distance, distance (signless) Laplacian integral.

Bicyclic graphs are connected graphs in which the number of edges equals the number of vertices plus one. Let P_p+1 = x₁x₂⋯x_p+1, P_t+1 = y₁y₂⋯y_t+1 and P_q+1 = z₁z₂⋯z_q+1 be three vertex-disjoint paths. Identifying the initial vertices as u₀ and the terminal vertices as v₀, the resultant graph, denoted by θ(p; t; q), is called a θ-graph. Let be the class of all bicyclic graphs on n vertices, which contain a θ-graph as an induced subgraph. In this paper, we study the distance spectral radius of bicyclic graphs in , and determine the graph with the largest distance spectral radius.

Trees are very common in the theory and applications of combinatorics. In this paper, we consider graphs whose underlying structure is a tree and study the behavior of the distance spectral radius under a graph transformation. As an application, we find the corona tree that maximizes the distance spectral radius among all corona trees with a fixed maximum degree. We also find the graph with minimal (maximal) distance spectral radius among all corona trees. Finally, we determine the graph with minimal distance spectral radius in a special class of corona trees.

Bicyclic graphs are connected graphs in which the number of edges equals the number of vertices plus one. The class of bicyclic graphs of order n, denoted by ℬ_n, can be partitioned into two subclasses: the class of graphs which contain induced ∞-graphs, and the class of graphs which contain induced θ-graphs. Bose et al. [2] have found the graph having the minimal distance spectral radius in . In this paper, we determine the graphs having the minimal distance spectral radius in . These results together give a complete characterization of the graphs having the minimal distance spectral radius in ℬ_n.

We characterize graphs with minimal distance spectral radius in two classes of graphs: with vertex connectivity k and minimum degree at least k, and with given number of blocks. Moreover, we determine the unique graph that maximizes the distance spectral radius among all graphs with given clique number.

Corona of two graphs has been defined in [F. Harary, Graph Theory (Addison-Wesley, 1969)]. In this paper, we study the distance and the distance Laplacian spectra of corona of two graphs and describe the complete distance (distance Laplacian) spectrum for some particular cases. As an application, we show that the corona operation can be used to create distance singular graphs. We also show that these results enable us to construct infinitely many pairs of distance (respectively, distance Laplacian) cospectral graphs. Last, we give a graph transformation and discuss its effect on the distance Laplacian spectral radius.

In this paper, we determine the unique tree that maximizes the distance spectral radius in the class of all trees in which each non-pendent vertex has degree at least $p$.

For a simple connected graph $G$ of order $n$ having distance Laplacian eigenvalues ${\rho }_1^L\ge {\rho }_2^L\ge \cdots \ge {\rho }_n^L$, the distance Laplacian energy $\mathrm{\text{DLE}}(G)$ is defined as $\mathrm{\text{DLE}}(G)={\sum }_{i=1}^n|{\rho }_i^L-\frac{2W(G)}{n}|$, where $W(G)$ is the Wiener index of $G$. We obtain the distance Laplacian spectrum of the joined union of graphs $G_1,G_2,\dots ,G_n$ in terms of their distance Laplacian spectrum and the spectrum of an auxiliary matrix. As application, we obtain the distance Laplacian spectrum of the lexicographic product of graphs. We study the distance Laplacian energy of connected graphs with given chromatic number $\chi$. We show that among all connected graphs with chromatic number $\chi$ the complete $\chi$-partite graph has the minimum distance Laplacian energy. Further, we discuss the distribution of distance Laplacian eigenvalues around average transmission degree $\frac{2W(G)}{n}$.

The distance energy of a connected graph $G$ is the sum of absolute values of the eigenvalues of the distance matrix of $G$. In this paper, we study how the distance energy of the complete split graph $GS(m,n)=K_m+{\overline{K}}_n$ changes when an edge is deleted from it. The complete split graph $GS(m,n)$ consists of a clique on $m$ vertices and an independent set on $n$ vertices in which each vertex of the clique is adjacent to each vertex of the independent set. We prove that the distance energy of the complete split graph $GS(m,n)$ always increases when an edge is deleted from it.

A conjugate skew gain graph is a skew gain graph with the labels (also called, the conjugate skew gains) from the field of complex numbers on the oriented edges such that they get conjugated when we reverse the orientation. In this paper, we introduce distance matrices for conjugate skew gain graphs and characterize balanced conjugate skew gain graphs using these matrices. We provide explicit formulae for the distance spectra of certain conjugate skew gain graphs.