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Eigenvalue bounds for some classes of matrices associated with graphs

Ranjit Mehatari, M. Rajesh Kannan (2021)

Czechoslovak Mathematical Journal

For a given complex square matrix A with constant row sum, we establish two new eigenvalue inclusion sets. Using these bounds, first, we derive bounds for the second largest and the smallest eigenvalues of adjacency matrices of k -regular graphs. Then we establish some bounds for the second largest and the smallest eigenvalues of the normalized adjacency matrices of graphs and the second smallest and the largest eigenvalues of the Laplacian matrices of graphs. The sharpness of these bounds is verified...

Eigenvalue Conditions for Induced Subgraphs

Jochen Harant, Julia Niebling, Sebastian Richter (2015)

Discussiones Mathematicae Graph Theory

Necessary conditions for an undirected graph G to contain a graph H as induced subgraph involving the smallest ordinary or the largest normalized Laplacian eigenvalue of G are presented.

Equienergetic graphs

Harishchandra S. Ramane, Hanumappa B. Walikar, Siddani Bhaskara Rao, B. Devadas Acharya, Prabhakar R. Hampiholi, Sudhir R. Jog, Ivan Gutman (2004)

Kragujevac Journal of Mathematics

Equienergetic self-complementary graphs

G. Indulal, A. Vijayakumar (2008)

Czechoslovak Mathematical Journal

In this paper equienergetic self-complementary graphs on p vertices for every p = 4 k , k ≥ 2 and p = 24 t + 1 , t ≥ 3 are constructed.

Exponents of two-colored digraphs

Yan Ling Shao, Yubin Gao (2009)

Czechoslovak Mathematical Journal

We consider the primitive two-colored digraphs whose uncolored digraph has n + s vertices and consists of one n -cycle and one ( n - 3 ) -cycle. We give bounds on the exponents and characterizations of extremal two-colored digraphs.

Extremal inverse eigenvalue problem for matrices described by a connected unicyclic graph

Bijoya Bardhan, Mausumi Sen, Debashish Sharma (2024)

Applications of Mathematics

In this paper, we deal with the construction of symmetric matrix whose corresponding graph is connected and unicyclic using some pre-assigned spectral data. Spectral data for the problem consist of the smallest and the largest eigenvalues of each leading principal submatrices. Inverse eigenvalue problem (IEP) with this set of spectral data is generally known as the extremal IEP. We use a standard scheme of labeling the vertices of the graph, which helps in getting a simple relation between the characteristic...

Extremal trees and molecular trees with respect to the Sombor-index-like graph invariants 𝒮𝒪 5 and 𝒮𝒪 6

Wei Gao (2024)

Czechoslovak Mathematical Journal

I. Gutman (2022) constructed six new graph invariants based on geometric parameters, and named them Sombor-index-like graph invariants, denoted by 𝒮𝒪 1 , 𝒮𝒪 2 , ⋯ , 𝒮𝒪 6 . Z. Tang, H. Deng (2022) and Z. Tang, Q. Li, H. Deng (2023) investigated the chemical applicability and extremal values of these Sombor-index-like graph invariants, and raised some open problems, see Z. Tang, Q. Li, H. Deng (2023). We consider the first open problem formulated at the end of Z. Tang, Q. Li, H. Deng (2023). We obtain the extremal values...

Extremal Unicyclic Graphs With Minimal Distance Spectral Radius

Hongyan Lu, Jing Luo, Zhongxun Zhu (2014)

Discussiones Mathematicae Graph Theory

The distance spectral radius ρ(G) of a graph G is the largest eigenvalue of the distance matrix D(G). Let U (n,m) be the class of unicyclic graphs of order n with given matching number m (m ≠ 3). In this paper, we determine the extremal unicyclic graph which has minimal distance spectral radius in U (n,m) Cn.

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