**Authors:** Omendra Singh, Pravin Garg, Neha Kansal

**Keywords:** Deficiency, maximum deficiency matrix, maxi-
mum deficiency eigenvalues, maximum deficiency energy.

### Abstract

The concept of maximum deficiency matrix $M_{df}(G)$ of a simple graph $G$ is introduced in this paper. Let $G=(V,E)$ be a simple graph of order $n$ and let $df(v_{i})$ be the deficiency of a vertex $v_{i}$, $i=1,2, \ldots, n$, then the maximum deficiency matrix $M_{df}(G) = [f_{ij}]_{n \times n}$ is defined as:
\[
f_{ij}=
\begin{cases}
max\{df(v_{i}), df(v_{j})\}, & \text{if $v_{i}v_{j} \in E(G)$} \\ 0~~~~~~~~~~~~~~~~~~~~~~~~, & \text{otherwise.} \end{cases}
\] Further, some coefficients of the characteristic polynomial $\phi(G; \gamma)$ of the maximum deficiency matrix of $G$ are obtained.
The maximum deficiency energy $E{M_{df}}(G)$ of a graph $G$ is also introduced. The bounds for $E{M_{df}}(G)$ are established. Moreover, maximum deficiency energy of some standard graphs is shown, and if the maximum deficiency energy of a graph is rational, then it must be an even integer.

Omendra Singh

Department of Mathematics,

University of Rajasthan,

Jaipur - 302004, India

Phone: +919521531401

E-mail:

Pravin Garg

Department of Mathematics,

University of Rajasthan,

Jaipur - 302004, India

Phone: +919982123280

E-mail:

Neha Kansal

Department of Mathematics,

University of Rajasthan,

Jaipur - 302004, India

Phone: +918952836930

E-mail:

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