Gossip algorithms are often considered suitable for wireless sensor networks (WSNs) because of their simplicity, fault tolerance, and adaptability to network changes. They are based on the idea of distributed information dissemination, where each node in the network periodically sends its information to randomly selected neighbors, leading to a rapid spread of information throughout the network. This approach helps reduce the communication overhead and ensures robustness against node failures. They have been commonly employed in WSNs owing to their low communication overheads and scalability. The time required for every node in the network to converge to the average of its initial value is called the average time. The average time is defined in terms of the second-largest eigenvalue of a stochastic matrix. Thus, estimating and analyzing the average time required for large-scale WSNs is computationally complex. This study derives explicit expressions of average time for WSNs and studies the effect of various network parameters such as communication link failures, topology changes, long-range links, network dimension, node transmission range, and network size. Our theoretical expressions substantially reduced the computational complexity of computing the average time to Oleft(n{-3}right). Furthermore, numerical results reveal that the long-range links and node transmission range of WSNs can significantly reduce average time, energy consumption, and absolute error for gossip algorithms.

Let G represent a simple connected graph with a vertex set denoted as { 1 , 2 , 3 , … , n} . The Laplacian matrix of G is denoted as L. The resistance distance of two vertices i and j is given by the expression: rij=lii†+ljj†-2lij†,i,j=1,2,3,⋯,n , where L†=(lij†)n×n represents the Moore–Penrose inverse of matrix L. The resistance matrix of the graph G is defined by R=(rij)n×n . Prior research in the literature has extensively explored determinants and inverses of the resistance matrix. This article extends the concept of a resistance matrix to incorporate matrix weights, particularly when dealing with symmetric matrix edge weights. It is demonstrated that the resistance matrix may lose its non-singularity property under this setup. The conditions for singularity of the resistance matrix R are established as necessary and sufficient. In cases where the resistance matrix becomes singular, the potential ranks of matrix R are characterized. An explicit equation is formulated to calculate the inverse and determinant of the resistance matrix in cases where it is not singular. Furthermore, for a specific scenario, the Moore–Penrose inverse of a singular resistance matrix is provided.

The distance, distance signless Laplacian and distance Laplacian matrix of a simple connected graph G, are denoted by D(G),DQ(G) = D(G) + Tr(G) and DL(G) = Tr(G) - D(G), respectively, where Tr(G) is the diagonal matrix of vertex transmission. Geršgorin discs for any n × n square matrix A = [aij] are the discs {z C: |z - aii|≤ Ri(A)}, where Ri(A) =σj/i|aij|,i = 1, 2,...,n. The famous Geršgorin disc theorem says that all the eigenvalues of a square matrix lie in the union of the Geršgorin discs of that matrix. In this paper, some classes of graphs are studied for which the smallest Geršgorin 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 Geršgorin 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 Geršgorin disc contains every distance Laplacian eigenvalue of the distance Laplacian matrix. These results give partial answers from some questions that are raised in [2].

A signed network ς = (G,σ) with the underlying graph G = (V,E), used as a mathematical model for analyzing social networks, has each edge in E with a weight 1 or-1 assigned by the signature function σ. In this paper, we deal with two types of Detour Distance Laplacian (DDL) matrices for signed networks. We characterize balance in signed social networks using these matrices and we compute the DDL spectrum of certain unbalanced signed networks, as balanced signed networks behave like unsigned ones.

Prism networks belong to generalized petersen graph topologies with planar and polyhedral properties. These networks are extensively used to study the complex networks in the context of computer science, biological networks, brain networks, and social networks. In this letter, the performance analysis of consensus algorithms over prism networks is investigated in terms of convergence time, network coherence, and communication delay. Specifically, in this letter, we first derive the explicit expressions for eigenvalues of Laplacian matrix over m -dimensional prism networks using spectral graph theory. Subsequently, the study of consensus metrics such as convergence time, maximum communication time delay, first order network coherence, and second order network coherence is performed. Our results indicate that the effect of noise and communication time-delay on the consensus dynamics in m -dimensional prism networks is minimal. The obtained results illustrate that the scale-free topology of m -dimensional prism networks along with loopy structure is responsible for strong robustness with respect to consensus dynamics in prism networks.

The distance matrix of a simple connected graph G is (Formula presented.) where dij is the length of a shortest path between the ith and jth vertices of G. Eigenvalues of D(G) are called the distance eigenvalues of G. The m-generation n-prism graph or (m, n)-prism graph can be defined in an iterative way where (Formula presented.) -prism graph is an n-vertex cycle. In this paper, we first find the number of zero eigenvalues of the distance matrix of a (m, n)-prism graph. Next, we find some quotient matrix that contains m nonzero distance eigenvalues of a (m, n)-prism graph. Our next result gives the rest of the nonzero distance eigenvalues of a (m, n)-prism graph in terms of distance eigenvalues of a cycle. Finally, we find the characteristic polynomial of the distance matrix of a (m, n)-prism graph. Applying this result, we provide the explicit distance eigenvalues of a (Formula presented.) -prism graph.

A dam is a man-made structural barrier built to store and control the flow of water in lakes or rivers. However, mismanagement of dams or extreme weather conditions lead to man-made or natural disasters respectively. Therefore, it is crucial to develop a efficient monitoring systems for maintaining a safe water levels in dams. In this work, we develop an Internet of Things(IoT) based monitoring system to the route the collected flood water dams automatically into the canal. In this system, water level is communicated to the base station using far-field communication. This information will be associated with the cloud and it is monitored by a command center. This command center can take the decision and can furnish the commands to lift the gates simultaneously. The ratio of the distribution of river water from dams to canals will be decided based on several aspects such as command area, water requirement, etc. An integrated system is developed using ARDUINO to meet the above requirements. This paper designs the efficient automated system for dams to effectively manage the water resources and prevent the man-made and natural calamities.

IoT contains a large amount of data as well as privacy sensitive information. This information can be easily identified by the various cyber-attackers. Attackers may steel the data or they can introduce viruses and other malicious software to damage the network. So to overcome these issues, we need to establish an efficient security mechanism for data protection in IoT systems. Advanced Encryption standard is one of the popular security protocols for IoT systems. The main disadvantage of this standard is power consumption for large sized IoT networks. In this paper we proposed an novel mechanism for secure communication in large sized IoT systems. In our design, we use a Hash based Message Authentication Code (HMAC) and a privacy-preserving communication protocol for chaos-based encryption. We observe that proposed mechanism consumes relatively less power over AES with less storage resources.

The distance matrix of a simple connected graph G is (Formula presented.), where (Formula presented.) is the distance between the vertices i and j in G. We consider a weighted tree T on n vertices with edge weights are square matrices of the same size. The distance (Formula presented.) between the vertices i and j is the sum of the weight matrices of the edges in the unique path from i to j. In this article, we establish a characterization for the trees in terms of the rank of (matrix) weighted Laplacian matrix associated with it. We present a necessary and sufficient condition for the distance matrix D, with matrix weights, to be invertible and the formula for the inverse of D, if it exists. Then we study some of the properties of the distance matrices of matrix weighted trees in connection with the Laplacian matrices, incidence matrices, and g-inverses. Finally, we derive an interlacing inequality for the eigenvalues of distance and Laplacian matrices for the case of positive definite matrix weights.

A partition of a square matrix A is said to be equitable if all the blocks of the partitioned matrix have constant row sums and each of the diagonal blocks are of square order. A quotient matrixQ of a square matrix A corresponding to an equitable partition is a matrix whose entries are the constant row sums of the corresponding blocks of A. A quotient matrix is a useful tool to find some eigenvalues of the matrix A. In this paper we determine some matrices whose eigenvalues are those eigenvalues of A which are not the eigenvalues of a quotient matrix of A. Using this result we find eigenvalue localization theorems for matrices having an equitable partition. In particular, we find eigenvalue localization theorems for stochastic matrices and give a suitable example to compare with the existing results.

The resistance matrix of a simple connected graph G is denoted by RG or simply by R and is defined by RG = (rij), where rij is the resistance distance between the vertices i and j in G. In this paper, we consider distance matrix of a weighted tree and the resistance matrix of any weighted graph, where the weights are nonzero scalars. We obtain the identities for minors of the Laplacian, resistance and distance matrices, which are independent of the nonsingularity of resistance and distance matrices. While finding these we obtain the necessary and sufficient condition for the resistance matrix to be singular and the rank of it. Finally, we obtain the Moore–Penrose inverse of R, when it is singular.

The resistance matrix of a simple connected graph G is denoted by R, and is defined by R=(r ij ), where r ij is the resistance distance between the vertices i and j of G. In this paper, we consider the resistance matrix of weighted graph with edge weights being positive definite matrices of same size. We derive a formula for the determinant and the inverse of the resistance matrix. Then, we establish an interlacing inequality for the eigenvalues of resistance and Laplacian matrices of tree. Using this interlacing inequality, we obtain the inertia of the resistance matrix of tree.

A positive eigenvector corresponding to the largest eigenvalue of a symmetric, non-negative and irreducible matrix is known as principal eigenvector of the matrix. For a real number p, 1 ≤ p< ∞, a principal eigenvector (y1,y2,…,yn)T is said to be p-norm normalized if (∑i=1nyip)1p=1. The distance matrix of a simple connected graph G is D(G) = (dij) , where dij is the distance between ith and jth vertices of G. The distance signless Laplacian matrix of the graph G is DQ(G) = D(G) + Tr(G) , where Tr(G) is a diagonal matrix whose ith diagonal entry is the transmission of the vertex i in G. In this paper we find some upper and lower bounds of the maximal and minimal entry of the p-normalized principal eigenvector for the distance matrix and distance signless Laplacian matrix of a graph and show that transmission regular graphs are extremal for all these bounds. We also compare these bounds with the bounds in the literature.

The power graph of a group G is a graph with vertex set G and two distinct vertices are adjacent if and only if one is an integral power of the other. In this paper we find both upper and lower bounds for the spectral radius of power graph of cyclic group Cn and characterize the graphs for which these bounds are extremal. Further we compute spectra of power graphs of dihedral group D2n and dicyclic group Q4n partially and give bounds for the spectral radii of these graphs.

The distance matrix of a simple connected graph G is D(G) = (dij), where dij is the distance between the ith and jth vertices of G. The distance signless Laplacian matrix of the graph G is DQ(G) = D(G) + Tr(G), where Tr(G) is a diagonal matrix whose ith diagonal entry is the transmission of the vertex i in G. In this paper, first, upper and lower bounds for the spectral radius of a nonnegative matrix are constructed. Applying this result, upper and lower bounds for the distance and distance signless Laplacian spectral radius of graphs are given, and the extremal graphs for these bounds are obtained. Also, upper bounds for the modulus of all distance (respectively, distance signless Laplacian) eigenvalues other than the distance (respectively, distance signless Laplacian) spectral radius of graphs are given. These bounds are probably first of their kind as the authors do not find in the literature any bound for these eigenvalues. Finally, for some classes of graphs, it is shown that all distance (respectively, distance signless Laplacian) eigenvalues other than the distance (respectively, distance signless Laplacian) spectral radius lie in the smallest Gerŝgorin disc of the distance (respectively, distance signless Laplacian) matrix.

The distance matrix of a simple graph G is D(G) = (di,j), where di,j is the distance between the ith and jth vertices of G. The distance spectral radius of G, written λ1(G), is the largest eigenvalue of D(G). We determine the distance spectral radius of the wheel graph Wn, a particular type of spider graphs, and the generalized Petersen graph P(n, k) for k ∈ {2, 3}.

The distance matrix of a simple graph G is D(G)=(^dij), where ^dij is the distance between ith and jth vertices of G. The spectrum of the distance matrix is known as the distance spectrum or D-spectrum of G. A simple connected graph G is called distance regular if it is regular, and if for any two vertices x,y ∈ G at distance i, there are constant number of neighbors ^ci and ^bi of y at distance i-1 and i+1 from x respectively. In this paper we prove that distance regular graphs with diameter d have at most d+1 distinct D-eigenvalues. We find an equitable partition and the corresponding quotient matrix of the distance regular graph which gives the distinct D-eigenvalues. We also prove that distance regular graphs satisfying ^bi=cd-<inf>1</inf> have at most ⌈d/2⌉ +2 distinct D-eigenvalues. Applying these results we find the distance spectrum of some distance regular graphs including the well known Johnson graphs. Finally we also answer the questions asked by Lin et al. [16].

The distance matrix of a simple connected graph G is D(G) = (dij), where dij is the distance between ith and jth vertices of G. The multiset of all eigenvalues of D(G) is known as the distance spectrum of G. Lin et al.(On the distance spectrum of graphs. Linear Algebra Appl., 439:1662-1669, 2013) asked for existence of graphs other than strongly regular graphs and some complete k-partite graphs having exactly three distinct distance eigenvalues. In this paper some classes of graphs with arbitrary diameter and satisfying this property is constructed. For each k ∈ {4, 5,⋯, 11} families of graphs that contain graphs of each diameter grater than k - 1 is constructed with the property that the distance matrix of each graph in the families has exactly k distinct eigenvalues. While making these constructions we have found the full distance spectrum of square of even cycles, square of hypercubes, corona of a transmission regular graph with K2, and strong product of an arbitrary graph with Kn.