University of Kashan Iranian Journal of Mathematical Chemistry 2228-6489 14 3 2023 09 01 Deficiency Sum Energy of Some Graph Classes 145 160 113971 10.22052/ijmc.2023.252633.1698 EN Omendra Singh Department of Mathematics, University of Rajasthan, Jaipur-302004, Rajasthan, India Pravin Garg Department of Mathematics, University of Rajasthan, Jaipur-302004, Rajasthan, India Neha Kansal Department of Mathematics, University of Rajasthan, Jaipur-302004, Rajasthan, India Journal Article 2023 03 09 In this paper‎, ‎we introduce the concept of deficiency sum matrix $S_{df}(G)$ of a simple graph $G=(V,E)$ of order $n$‎. ‎The deficiency $df(v)$ of a vertex $v \in V$ is the deviation between the degree of the vertex $v$‎<br />‎ and the maximum degree of the graph‎. ‎The deficiency sum matrix $S_{df}(G)$ is a matrix of order $n$ whose $(i,j)$-th entry is $df(v_{i})+df(v_{j})$‎, ‎if the vertices $v_{i}$ and $v_{j}$ are adjacent and $0$‎, ‎otherwise‎. ‎In addition‎, ‎we introduce deficiency sum energy $ES_{df}(G)$ of a graph $G$ and establish some bounds for $ES_{df}(G)$‎. ‎Further‎, ‎deficiency sum energy of some classes of graphs are obtained‎. ‎Moreover‎, ‎we construct an algorithm and python(3.8) code to find out spectrum and deficiency sum energy of graph $G$‎. https://ijmc.kashanu.ac.ir/article_113971_6a0f0a71de5939bc3635ff1461755abd.pdf