On spectral properties of neighbourhood second Zagreb matrix of graph

Document Type : Original paper

Authors

School of Basic Sciences, IIT Bhubaneswar, Bhubaneswar, 752050, India

Abstract

Let G be a simple graph with vertex set V(G)={1,2,…,n} and δ(i)=∑{i,j}∈E(G)d(j), where d(j) is the degree of the vertex j in G. Inspired by the second Zagreb matrix and neighborhood first Zagreb matrix of a graph, we introduce the neighborhood second Zagreb matrix of G, denoted by NF(G). It is the n×n matrix whose ij-th entry is equal to δ(i)δ(j), if i and j are adjacent in G and 0, otherwise. The neighborhood second Zagreb spectral radius ρNF(G) is the largest eigenvalue of NF(G). The neighborhood second Zagreb energy E(NF) of the graph G is the sum of the absolute values of the eigenvalues of NF(G). In this paper, we obtain some spectral properties of NF(G). We provide sharp bounds for ρNF(G) and E(NF), and obtain the corresponding extremal graphs.

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References

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Communications in Combinatorics and Optimization
Volume 10, Issue 2
June 2025
Pages 275-293

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