## 1 Introduction

All graphs considered in this paper are finite, simple and undirected. In particular, these graphs do not have loops. Let *G* = (*V*, *E*) be a graph with the vertex set *V*(*G*) = {*v*_{1}, *v*_{2}, *v*_{3}, ⋯, *v _{n}*} and the edge set

*E*(

*G*) = {

*e*

_{1},

*e*

_{2},

*e*

_{3}, ⋯,

*e*}, that is |

_{m}*V*(

*G*)| =

*n*and |

*E*(

*G*)| =

*m*. The vertex

*u*and

*v*are adjacent if

*uv*∈

*E*(

*G*). The open(closed) neighborhood of a vertex

*v*∈

*V*(

*G*) is

*N*(

*v*) = {

*u*:

*uv*∈

*E*(

*G*)} and

*N*[

*v*] =

*N*(

*v*) ∪ {

*v*} respectively. The degree of a vertex

*v*∈

*V*(

*G*) is denoted by

*d*(

_{G}*v*) and is defined as

*d*(

_{G}*v*) = |

*N*(

*v*)|. A vertex

*v*∈

*V*(

*G*) is pendant if |

*N*(

*v*)| = 1 and is called supporting vertex if it is adjacent to pendant vertex. Any vertex

*v*∈

*V*(

*G*) with |

*N*(

*v*)| > 1 is called internal vertex. If

*d*(

_{G}*v*) =

*r*for every vertex

*v*∈

*V*(

*G*), where

*r*∈ ℤ

^{+}then

*G*is called

*r*-regular. If

*r*= 2 then it is called cycle graph

*C*and for

_{n}*r*= 3 it is called the cubic graph. A graph

*G*is unicyclic if |

*V*| = |

*E*|. A graph

*G*is called a block graph, if every block in

*G*is a complete graph. For undefined terminologies we refer the reader to [16].

A subset *D* ⊆ *V*(*G*) is called dominating set if *N*[*D*] = *V*(*G*). The minimum cardinality of such a set *D* is called the domination number *γ*(*G*) of *G*. A dominating set *D* is connected if the subgraph induced by *D* is connected. The minimum cardinality of connected dominating set *D* is called the connected dominating number *γ _{c}*(

*G*) of

*G*[27].

The energy *E*(*G*) of a graph *G* is equal to the sum of the absolute values of the eigenvalues of the adjacency matrix of *G*. This quantity, introduced almost 30 years ago [13] and having a clear connection to chemical problems [15], has in newer times attracted much attention of mathematicians and mathematical chemists [3, 8, 9, 10, 11, 12, 20,22, 23, 24, 28, 30, 31].

In connection with energy (that is defined in terms of the eigenvalues of the adjacency matrix), energy-like quantities were considered also for the other matrices: Laplacian [15], distance [17], incidence [18], minimum covering energy [1] etc. Recall that a great variety of matrices has so far been associated with graphs [4, 5, 10, 29].

Recently in [25] the authors have studied the dominating matrix which is defined as:

Let *G* = (*V*, *E*) be a graph with *V*(*G*) = {*v*_{1}, *v*_{2}, ⋯, *v _{n}*} and let

*D*⊆

*V*(

*G*) be a minimum dominating set of

*G*. The minimum dominating matrix of

*G*is the

*n*×

*n*matrix defined by

*A*(

_{D}*G*) = (

*a*), where

_{ij}*a*= 1 if

_{ij}*v*∈

_{i}v_{j}*E*(

*G*) or

*v*=

_{i}*v*∈

_{j}*D*, and

*a*= 0 if

_{ij}*v*∉

_{i}v_{j}*E*(

*G*).

The characteristic polynomial of *A _{D}*(

*G*) is denoted by

*f*(

_{n}*G*,

*μ*) :=

*det*(

*μI*–

*A*(

_{D}*G*)).

The minimum dominating eigenvalues of a graph *G* are the eigenvalues of *A _{D}*(

*G*). Since

*A*(

_{D}*G*) is real and symmetric, its eigenvalues are real numbers and we label them in non-increasing order

*μ*

_{1}≥

*μ*

_{2}≥ ⋯ ≥

*μ*. The minimum dominating energy of

_{n}*G*is then defined as

Motivated by dominating matrix, here we define the minimum connected dominating matrix abbreviated as (c-dominating matrix). The c-dominating matrix of *G* is the *n* × *n* matrix defined by *A _{Dc}*(

*G*) = (

*a*), where

_{ij}The characteristic polynomial of *A _{Dc}*(

*G*) is denoted by

*f*(

_{n}*G*,

*λ*) :=

*det*(

*λ I*–

*A*(

_{Dc}*G*)).

The c-dominating eigenvalues of a graph *G* are the eigenvalues of *A _{Dc}*(

*G*). Since

*A*(

_{Dc}*G*) is real and symmetric, its eigenvalues are real numbers and we label them in non-increasing order

*λ*

_{1}≥

*λ*

_{2}≥ ⋯ ≥

*λ*. The c-dominating energy of

_{n}*G*is then defined as

To illustrate this, consider the following examples:

Let *G* be the 5-vertex path *P*_{5}, with vertices *v*_{1}, *v*_{2}, *v*_{3}, *v*_{4}, *v*_{5} and let its minimum connected dominating set be *D _{c}* = {

*v*

_{2},

*v*

_{3},

*v*

_{4}}. Then

The characteristic polynomial of *A _{Dc}*(

*G*) is

*λ*

^{5}– 3

*λ*

^{4}–

*λ*

^{3}+ 5

*λ*

^{2}+

*λ*– 1 = 0. The minimum connected dominating eigenvalues are

*λ*

_{1}= 2.618,

*λ*

_{2}= 1.618,

*λ*

_{3}= 0.382,

*λ*

_{4}= –1.000 and

*λ*

_{5}= –0.618.

Therefore, the minimum connected dominating energy is *E _{Dc}*(

*P*

_{5}) = 6.236.

Consider the following graph

Let *G* be a tree *T* as shown above and let its minimum connected dominating set be *D _{c}* = {

*b*,

*d*,

*e*,

*f*,

*g*,

*h*}. Then

By direct calculation, we get the minimum connected dominating eigenvalues are *λ*_{1} = 2.945, *λ*_{2} = 2.596, *λ*_{3} = 1.896, *λ*_{4} = 1.183, *λ*_{5} = –1.263, *λ*_{6} = –1.152, *λ*_{7} = 0.579, *λ*_{8} = 0.000, *λ*_{9} = –0.268 and *λ*_{10} = –0.516.

Therefore, the minimum connected dominating energy is *E _{Dc}*(

*T*) = 12.398.

The c-dominating energy of the following graphs can be calculated easily:

*E*(_{Dc}*K*) = (_{n}*n*– 2) + , where$\begin{array}{}\sqrt{n(n-2)+5}\end{array}$ *K*is the complete graph of order_{n}*n*.*E*(_{Dc}*K*_{1,n–1}) = where$\begin{array}{}\sqrt{4n-3}\end{array}$ *K*_{1,n–1}is the star graph.*E*(_{Dc}*K*_{n×2}) = (2*n*– 3) + , where$\begin{array}{}\sqrt{4n(n-1)-9}\end{array}$ *K*_{n×2}is the coctail party graph.

In this paper, we are interested in studying the mathematical aspects of the c-dominating energy of a graph. This paper has organized as follows: The section 1, contains the basic definitions and background of the current topic. In section 2, we show the chemical applicability of c-dominating energy for molecular graphs *G*. The section 3, contains the mathematical properties of c-dominating energy. In the last section, we have characterized, trees, unicyclic graphs and cubic graphs and block graphs with equal minimum dominating energy and c-dominating energy. Finally, we conclude this paper by posing an open problem.

## 2 Chemical Applicability of *E*_{Dc}(*G*)

_{Dc}

We have used the c-dominating energy for modeling eight representative physical properties like boiling points(bp), molar volumes(mv) at 20^{∘}*C*, molar refractions(mr) at 20^{∘}*C*, heats of vaporization (hv) at 25^{∘}*C*, critical temperatures(ct), critical pressure(cp) and surface tension (st) at 20^{∘}*C* of the 74 alkanes from ethane to nonanes. Values for these properties were taken from http://www.moleculardescriptors.eu/dataset.htm. The c-dominating energy *E _{Dc}*(

*G*) was correlated with each of these properties and surprisingly, we can see that the

*E*has a good correlation with the heats of vaporization of alkanes with correlation coefficient

_{Dc}*r*= 0.995.

The following structure-property relationship model has been developed for the c-dominating energy *E _{Dc}*(

*G*).

## 3 Mathematical Properties of c-Dominating Energy of Graph

We begin with the following straightforward observations.

*Note that the trace of**A _{Dc}*(

*G*) =

*γ*(

_{c}*G*).

*Let**G* = (*V*, *E*) *be a graph with**γ _{c}*-

*set*

*D*.

_{c}*Let*

*f*(

_{n}*G*,

*λ*) =

*c*

_{0}

*λ*+

^{n}*c*

_{1}

*λ*

^{n–1}+ ⋯ +

*c*

_{n}*be the characteristic polynomial of*

*G*.

*Then*

*c*_{0}= 1,*c*_{1}= –|*D*| = –_{c}*γ*(_{c}*G*).

*If**λ*_{1}, *λ*_{2}, ⋯, *λ _{n}*

*are the eigenvalues of*

*A*(

_{Dc}*G*),

*then*

$\begin{array}{}{\displaystyle \sum _{i=1}^{n}{\lambda}_{i}={\gamma}_{c}(G)}\end{array}$ $\begin{array}{}{\displaystyle \sum _{i=1}^{n}{\lambda}_{i}^{2}=2m+{\gamma}_{c}(G).}\end{array}$

- Follows from Observation 1.
- The sum of squares of the eigenvalues of
*A*(_{Dc}*G*) is just the trace of*A*(_{Dc}*G*)^{2}. Therefore□$$\begin{array}{}\begin{array}{rll}\sum _{i=1}^{n}{\lambda}_{i}^{2}& =& \sum _{i=1}^{n}\sum _{j=1}^{n}{a}_{ij}{a}_{ji}\\ & =& 2\sum _{i<j}({a}_{ij}{)}^{2}+\sum _{i=1}^{n}({a}_{ii}{)}^{2}\\ & =& 2m+{\gamma}_{c}(G).\end{array}\end{array}$$

We now obtain bounds for *E _{Dc}*(

*G*) of

*G*, similar to McClelland’s inequalities [21] for graph energy.

*Let**G**be a graph of order**n**and size**m**with**γ _{c}*(

*G*) =

*k*.

*Then*

Let *λ*_{1} ≥ *λ*_{2} ≥ ⋯ ≥ *λ _{n}* be the eigenvalues of

*A*(

_{Dc}*G*). Bearing in mind the Cauchy-Schwarz inequality,

we choose *a _{i}* = 1 and

*b*= |

_{i}*λ*|, which by Theorem 3 implies

_{i}□

*Let**G**be a graph of order**n**and size**m**with**γ _{c}*(

*G*) =

*k*.

*Let*

*λ*

_{1}≥

*λ*

_{2}≥ ⋯ ≥

*λ*

_{n}*be a non-increasing arrangement of eigenvalues of*

*A*(

_{Dc}*G*).

*Then*

*where**where* [*x*] *denotes the integer part of a real number**k*.

Let *a*_{1}, *a*_{2}, ⋯, *a _{n}* and

*b*

_{1},

*b*

_{2}, ⋯,

*b*be real numbers for which there exist real constants

_{n}*a*,

*b*,

*A*and

*B*, so that for each

*i*,

*i*= 1, 2, ⋯,

*n*,

*a*≤

*a*≤

_{i}*A*and

*b*≤

*b*≤

_{i}*B*. Then the following inequality is valid (see [6]).

where *a*_{1} = *a*_{2} = ⋯ = *a _{n}* and

*b*

_{1}=

*b*

_{2}= ⋯ =

*b*.

_{n}We choose *a _{i}* := |

*λ*|,

_{i}*b*:= |

_{i}*λ*|,

_{i}*a*=

*b*:= |

*λ*| and

_{n}*A*=

*B*:= |

*λ*

_{1}|,

*i*= 1, 2, ⋯,

*n*, inequality (4) becomes

Since

Hence equality holds if and only if *λ*_{1} = *λ*_{2} = ⋯ = *λ _{n}*.□

*Let**G**be a graph of order**n**and size**m**with**γ _{c}*(

*G*) =

*k*.

*Let*

*λ*

_{1}≥

*λ*

_{2}≥ ⋯ ≥

*λ*

_{n}*be a non*-

*increasing arrangement of eigenvalues of*

*A*(

_{Dc}*G*).

*Then*

Since

*Let**G**be a graph of order**n**and size**m**with**γ _{c}*(

*G*) =

*k*.

*Let*

*λ*

_{1}≥

*λ*

_{2}≥ ⋯ ≥

*λ*

_{n}*be a non-increasing arrangement of eigenvalues of*

*A*(

_{Dc}*G*).

*Then*

Let *a*_{1}, *a*_{2}, ⋯, *a _{n}* and

*b*

_{1},

*b*

_{2}, ⋯,

*b*be real numbers for which there exist real constants

_{n}*r*and

*R*so that for each

*i*,

*i*= 1, 2, ⋯,

*n*holds

*ra*≤

_{i}*b*≤

_{i}*Ra*. Then the following inequality is valid (see [11]).

_{i}Equality of (8) holds if and only if, for at least one *i*, 1 ≤ *i* ≤ *n* holds *ra _{i}* =

*b*=

_{i}*Ra*.

_{i}For *b _{i}* := |

*λ*|,

_{i}*a*:= 1

_{i}*r*:= |

*λ*| and

_{n}*R*:= |

*λ*

_{1}|,

*i*= 1, 2, ⋯,

*n*inequality (8) becomes

Since

Hence the result.□

*Let**G**be a graph of order**n**and size**m**with**γ _{c}*(

*G*) =

*k*.

*If*

*ξ*= |

*detA*(

_{Dc}*G*)|,

*then*

Employing the inequality between the arithmetic and geometric means, we obtain

Thus,

□

*If**λ*_{1}(*G*) *is the largest minimum connected dominating eigenvalue of**A _{Dc}*(

*G*),

*then*

Let *X* be any non-zero vector. Then we have *λ*_{1}(*A _{Dc}*(

*G*)) ≥

Next, we obtain Koolen and Moulton’s [19] type inequality for *E _{Dc}*(

*G*).

*If**G**is a graph of order**n**and size**m**and* 2*m* + *γ _{C}*(

*G*) ≥

*n*,

*then*

Bearing in mind the Cauchy-Schwarz inequality,

Put *a _{i}* = 1 and

*b*= |

_{i}*λ*| then

_{i}Let

For decreasing function

Since (2*m* + *k*) ≥ *n*, we have

Hence by (12), the result follows.□

## 4 Graphs with equal Dominating and c-Dominating Energy

Its a natural question to ask that for which graphs the dominating energy and c-dominating energy are equal. To answer this question, we characterize graphs with equal dominating energy and c-dominating energy. The graphs considered in this section are trees, cubic graphs, unicyclic graphs, block graphs and cactus graphs.

*Let**G* = *T**be a tree with at least three vertices*, *then**E _{D}*(

*G*) =

*E*(

_{Dc}*G*)

*if and only if every internal vertex of*

*T*

*is a support vertex*.

Let *G* = *T* be a tree of order at least 3. Let *F* = {*u*_{1}, *u*_{2}, ⋯, *u _{k}*} be the set of internal vertices of

*T*. Then clearly

*F*is the minimal dominating set of

*G*. Therefore in

*A*(

_{D}*G*) the values of

*u*= 1 in the diagonal entries. Further, observe that 〈

_{i}*F*〉 is connected. Hence

*F*is the minimal connected dominating set. Therefore,

*A*(

_{D}*G*) =

*A*(

_{Dc}*G*). In general,

*A*(

_{D}*G*) =

*A*(

_{Dc}*G*) is true if every minimum dominating set is connected. In other words,

*A*(

_{D}*G*) =

*A*(

_{Dc}*G*) if

*γ*(

*G*) =

*γ*(

_{c}*G*). Therefore, the result follows from Theorem 2.1 in [2].□

In the next three theorems we characterize unicyclic graphs with *A _{D}*(

*G*) =

*A*(

_{Dc}*G*). Since,

*A*(

_{D}*G*) =

*A*(

_{Dc}*G*) if

*γ*(

*G*) =

*γ*(

_{c}*G*). Therefore, the proof of our next three results follows from Theorem 2.2, Theorem 2.4 and Theorem 2.5 in [2].

*Let**G**be a unicyclic graph with cycle**C* = *u*_{1}*u*_{2} ⋯, *u _{n}*

*u*

_{1}

*n*≥ 5

*and let*

*X*= {

*v*∈

*C*:

*d*(

_{G}*v*) ≥ 2}.

*Then*

*E*(

_{D}*G*) =

*E*(

_{Dc}*G*)

*if the following conditions hold*:

*(a). Every**v*∈*V*–*N*[*X*]*with**d*(_{G}*V*) ≥ 2*is a support vertex*.*(b)*. 〈*X*〉*is connected and*|*X*| ≤ 3.*(c). If*〈*X*〉 =*P*_{1}*or**P*_{3},*both vertices in**N*(*X*)*of degree at least 3 are supports and if*〈*X*〉 =*P*_{2},*at least one vertex in**N*(*X*)*of degree at least three is a support*.

*Let**G**be unicyclic graph with* |*V*(*G*)| ≥ 4 *containing a cycle**C* = *C*_{3}, *and let**X* = {*v* ∈ *C* : *d _{G}*(

*v*) = 2}.

*Then*

*E*(

_{D}*G*) =

*E*(

_{Dc}*G*)

*if the following conditions hold*:

*(a). Every**v*∈*V*–*N*[*X*]*with**d*(_{G}*V*) ≥ 2*is a support vertex*.*(b). There exists some unique**v*∈*C**with**d*(_{G}*v*) ≥ 3*or for every**v*∈*C**of**d*(_{G}*v*) ≥ 3*is a support*.

*Let**G**be unicyclic graph with* |*V*(*G*)| ≥ 5 *containing a cycle**C* = *C*_{4}, *and let**X* = {*v* ∈ *C* : *d _{G}*(

*v*) = 2}.

*Then*

*E*(

_{D}*G*) =

*E*(

_{Dc}*G*)

*if the following conditions hold*:

*(a). Every**v*∈*V*–*N*[*X*]*with**d*(_{G}*V*) ≥ 2*is a support vertex*.*(b). If*|*X*| = 1,*all the three remaining vertices of**C**are supports and if*|*X*| ≥ 2,*C**contains at least one support*.

*Let**G**be a connected cubic graph of order**n*, *Then**E _{D}*(

*G*) =

*E*(

_{Dc}*G*)

*if*

*G*≅

*K*

_{4},

*C*

_{6},

*K*

_{3,3}

*G*

_{1}

*or*

*G*

_{2}

*where*

*G*

_{1}

*and*

*G*

_{2}

*are given in Fig. 4*.

*Let**G**be a block graph of with**l* ≥ 2. *Then**E _{D}*(

*G*) =

*E*(

_{Dc}*G*)

*if every cutvertex of*

*G*

*is an end block cutvertex*.

Since *E _{D}*(

*G*) =

*E*(

_{Dc}*G*) if

*γ*(

*G*) =

*γ*(

_{c}*G*). Therefore, the result follows from Theorem 2 in [7].□

We conclude this paper by posing the following open problem for the researchers:

**Open Problem**: Construct non- cospectral graphs with unequal domination and connected domination numbers having equal dominating energy and c-dominating energy.

^{}

Communicated by Juan Luis García Guirao

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