Related papers: Two new topological indices based on graph adjacen…
We present sharp inequalities relating the number of vertices, edges, and triangles of a graph to the smallest eigenvalue of its adjacency matrix and the largest eigenvalue of its Laplacian.
In this work we perform a detailed statistical analysis of topological and spectral properties of random geometric graphs (RGGs); a graph model used to study the structure and dynamics of complex systems embedded in a two dimensional space.…
Twin vertices of a graph have the same open neighbourhood. If they are not adjacent, then they are called duplicates and contribute the eigenvalue zero to the adjacency matrix. Otherwise they are termed co-duplicates, when they contribute…
Topological indices have important role in theoretical chemistry for QSPR researches. Among the all topological indices the Randi\'c and the Zagreb indices have been used more considerably than any other topological indices in chemical and…
We give inequalities relating the eigenvalues of the adjacency matrix and the Laplacian of a graph, and its minimum and maximum degrees. The results are applied to derive new conditions for quasi-randomness of graphs.
An eigenvalue of the adjacency matrix of a graph is said to be \emph{main} if the all-1 vector is not orthogonal to the associated eigenspace. In this work, we approach the main eigenvalues of some graphs. The graphs with exactly two main…
Suppose $G$ is a simple graph on $n$ vertices. The $D$-eigenvalues $\mu_1,\mu_2,\cdots,\mu_n$ of $G$ are the eigenvalues of its distance matrix. The distance Estrada index of $G$ is defined as $DEE(G)=\sum_{i=1}^ne^{\mu_i}$. In this paper,…
Many real-world networks exhibit a high degeneracy at few eigenvalues. We show that a simple transformation of the network's adjacency matrix provides an understanding of the origins of occurrence of high multiplicities in the networks…
Typically, graph structures are represented by one of three different matrices: the adjacency matrix, the unnormalised and the normalised graph Laplacian matrices. The spectral (eigenvalue) properties of these different matrices are…
The anti-adjacency matrix of a graph is constructed from the distance matrix of a graph by keeping each row and each column only the largest distances. This matrix can be interpreted as the opposite of the adjacency matrix, which is instead…
We consider topological indices I that are sums of f(deg(u)) f(deg(v)), where {u,v} are adjacent vertices and f is a function. The Randi{\'c} connectivity index or the Zagreb group index are examples for indices of this kind. In earlier…
Given a simple graph $G$, its $A_\alpha$ matrix is a convex combination with parameter $\alpha\in [0,1]$ of its adjacency matrix and its degree diagonal matrices. Here we compare two lower bounds presented in [J. D. G. Silva Jr., C. S.…
We analyse the eigenvectors of the adjacency matrix of a random inhomogeneous graph constructed from a specified degree sequence. We assume that the empirical degree sequence has bounded mean and variance. We show that near the edges of the…
Topological indices are molecular descriptors that describe the properties of chemical compounds. These topological indices correlate specific physico-chemical properties like boiling point, enthalpy of vaporization, strain energy, and…
In this paper, we give a new proof on a Theorem of Tutte which says that the determinants of the adjacency matrices of two hypomorphic graphs are the same. We also study the lowest eigenvectors.
Given a connected graph, the principal eigenvector of the adjacency matrix (often called the Perron vector) can be used to assign positive weights to the vertices. A natural way to measure the homogeneousness of this vector is by…
In this paper, we discuss various connections between the smallest eigenvalue of the adjacency matrix of a graph and its structure. There are several techniques for obtaining upper bounds on the smallest eigenvalue, and some of them are…
Centrality represents a fundamental research field in complex network analysis, where centrality measures identify important vertices within networks. Over the years, researchers have developed diverse centrality measures from varied…
In this paper, four novel topological indices named as neighbourhood version of forgotten topological index (F_N), modified neighbourhood version of Forgotten topological index ($F_N^*$), neighbourhood version of second Zagreb index…
Graph autoencoders (GAEs) are powerful tools in representation learning for graph embedding. However, the performance of GAEs is very dependent on the quality of the graph structure, i.e., of the adjacency matrix. In other words, GAEs would…