Published in Special Matrices
In this article, we investigate connected signed graphs which have a connected star complement for both −2-2 and 2 (i.e. simultaneously for the two eigenvalues), where −2-2 (resp. 2) is the least (largest) eigenvalue of the adjacency matrix of a signed graph under consideration. We determine all such star complements and their maximal extensions (a...
Published in Discussiones Mathematicae Graph Theory
For any S ⊂ ℤ we say that a graph G has the S-property if there exists an S-edge-weighting w : E(G) → S such that for any pair of adjacent vertices u, v we have ∑e∈E(v) w(e) ≠ ∑e∈E(u) w(e), where E(v) and E(u) are the sets of edges incident to v and u, respectively. This work focuses on {a, a + 2}-edge-weightings where a ∈ ℤ is odd. We show that a ...
Published in Acta Universitatis Sapientiae, Informatica
Let R be a commutative ring with unity 1 ≠ 0 and let R× be the set of all unit elements of R. The unitary Cayley graph of R, denoted by GR = Cay(R, R×), is a simple graph whose vertex set is R and there is an edge between two distinct vertices x and y of R if and only if x − y ∈ R×. In this paper, we determine the Laplacian and signless Laplacian e...
Published in Journal of Applied Mathematics, Statistics and Informatics
With this article in mind, we have found some results using eigenvalues of graph with sign. It is intriguing to note that these results help us to find the determinant of Normalized Laplacian matrix of signed graph and their coe cients of characteristic polynomial using the number of vertices. Also we found bounds for the lowest value of eigenvalue...
Published in Computational and Applied Mathematics
In this paper, we focus on the index (largest eigenvalue) of the adjacency matrix of connected signed graphs. We give some general results on the index when the corresponding signed graph is perturbed. As applications, we determine the first five largest indices among all unbalanced signed bicyclic graphs on n≥36\documentclass[12pt]{minimal} \usepa...
Published in Computational and Applied Mathematics
In this paper, we consider a generalized join operation, the H˙\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{H}$$\end{document}-join on signed graphs, where H˙\d...
Published in Special Matrices
We derive an inequality that includes the largest eigenvalue of the adjacency matrix and walks of an arbitrary length of a signed graph. We also consider certain particular cases.
The Estrada index of a graph/network is defined as the trace of the adjacency matrix exponential. It has been extended to other graph-theoretic matrices, such as the Laplacian, distance, Seidel adjacency, Harary, etc. Here, we describe many of these extensions, including new ones, such as Gaussian and Mittag-Leffler ones. More importantly, we conte...
Published in Acta Universitatis Sapientiae, Mathematica
A signed graph Σ is a graph with positive or negative signs attatched to each of its edges. A signed graph Σ is balanced if each of its cycles has an even number of negative edges. Restrained dominating set D in Σ is a restrained dominating set of its underlying graph where the subgraph induced by the edges across Σ[D : V \ D] and within V \ D is b...
Published in Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
In this note, we generalize an interesting recent characterization of tridiagonal M-matrices whose inverse is still tridiagonal to any acyclic matrix. We discuss a possible extension. Several illustrative examples are provided.
