# Energy of Certain Classes of Graphs Determined by Their Laplacian Degree Product Adjacency Spectrum

Authors: Asim Khurshid; Muhammad Salman; Masood Ur Rehman; Mohammad Tariq Rahim
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this study, we investigate the Laplacian degree product spectrum and corresponding energy of four families of graphs, namely, complete graphs, complete bipartite graphs, friendship graphs, and corona products of 3 and 4 cycles with a null graph.


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## Body

## 1. Introduction

The graph energy was firstly introduced by Ivan Gutman in 1978 [1]. His idea was motivated by the well-known Hückel molecular orbital theory by Erich Hückel in 1930s, which permits pharmacologists to imprecise energies associated π-electron orbital of molecules called conjugated hydrocarbons [2]. The spectrum and the energy of a graph have significant applications and connections in the branches of Mathematics, such as linear algebra and combinatorial optimization fields which have lot to do with graph spectrum and energy. The combinatorial and graph theoretical approaches have strong bonding to solve real-life problems. Many results and methods from the spectral graph theory can be applied for the practicalities and evolution of matrix theory [3]. An ordered pair Γ=V,E, called a graph with vertex set of Γ, is denoted by V and its edge set by E. Two vertices u,v are adjacent if they make an edge in Γ, and we denote it by u∼v. The number of edges incident to a vertex v of Γ is the degree of v, and it is denoted by dv [4, 5]. The adjacency matrix of Γ, of order n denoted by AΓ, is a square symmetric matrix of order n×n whose ijth element can be found as [3](1)aij=0,ifu≁v,numberofedgesbetweenuandv,ifu∼v.For energy and spectrum of graphΓ, let AΓ be the adjacency matrix, the summation of absolute values of its eigenvalues compose energy of graph and these eigenvalues related with their multiplicities forms the spectrum of graph [4], i.e.,(2)SpΓ=λ1λ2…λnnλ1nλ2…nλn,and(3)EΓ=∑i=1nλi,where nλ1,nλ2,…,nλn are the multiplicities of the eigenvalues λ1,λ2,…,λn of AΓ. In [6], the degree product adjacency matrix, for a simple connected graph Γ having n vertices say v1,v2,…,vn, is a real symmetric matrix, denoted by DPAΓ=dij,1 with(4)dij=dvidvj,ifvi∼vj,0,otherwise.The Laplacian degree product adjacency matrix ofΓ is defined as(5)LDPAΓ=DPAΓ−DΓ,where DΓ is the degree matrix of Γ having diagonal entries as the degree of each vertex and all other entries are zero. The spectrum (1) and energy (2) obtained correspond to the eigenvalues of LDPAΓ and are called the Laplacian degree product adjacency spectrum and energy, LSpDPAΓ and LEDPAΓ, respectively [7], as the degree sum concept was conceived earlier in [8].


## 2. Main Results

In this module, we study the Laplacian degree product adjacency spectrum and energy of some well-known families of graphs, such as complete graphs, complete bipartite graphs, friendship graphs, and corona products of 3 and 4 cycles with null graph. We also evaluate the correct spectrum and the energy of degree product adjacency matrix of the corona product of 4 cycle with null graphs (thorny 4-cycle ring), which was found incorrect in [6].


### 2.1. Complete GraphsKx

Letv1,v2…,vx be the vertex set of Kx; then, the following result provides the Laplacian degree product adjacency spectrum and energy of Kx.Theorem 1.
Forx≥2, let Kx be a complete graph. Then,(6)LSpDPAKx=xx2−3x+2x1−x1x−1,and Laplacian degree product adjacency energy of Kx is 22x−3-times the size of Kx.Proof.
First of all note thatdvi=x−1, for each 1≤i≤x. Accordingly, we have the following Laplacian degree product adjacency matrix:(7)LDPAKx=v1v2v3…vxv1v2v3…vx1−x1−x21−x2…1−x21−x21−x1−x2…1−x21−x21−x21−x…1−x2⋮⋮⋮⋱⋮1−x21−x21−x2…1−x.
Eigenvalues ofLDPAKx are(8)x1−x,x−1−times,xx2−3x+2,1−time.
These eigenvalues provide the required spectrum. Moreover, by (3), we have(9)LEDPAKx=x−1x1−x+xx2−3x+2=22x−3x2.
Since the size ofKx is x2, so the result is proved.


### 2.2. Complete Bipartite GraphsKx,y

Let a complete bipartite graphKx,y with vertex sets VxΓ=v1,v2,…,vx and VyΓ=vx+1,vx+2,…,vy be as partitions. The order and the size of Kx,y graph are x+y and xy, respectively. Then, the Laplacian degree product adjacency spectrum and energy of Kx,y can be obtained from the following result.Theorem 2.
Forx,y≥1, a complete bipartite graph Kx,y, then(10)LSpDPAKx,y=−x+y2±124y3x3+1−x−y1y−1x−1,y=x+1withx≥1,−x+y2±124y3x3+y−x2−x−y1y−1x−1,y>x≥1withy≠x+1,xx2−1−xx2+1−x112x−1,x=y≥1,−1+y2±124y3+1−y2−11y−1x=1andy≥1.
Moreover,(11)LEDPAKx,y=2x3+x2−x,wheneverx=y≥1,2xy,otherwise.Proof.
Note thatdvi=y, for each 1≤i≤x and dvj=x, for each x+1≤j≤y. Accordingly, the Laplacian degree product adjacency matrix of Kx,y is(12)LDPAKx,y=v1v2v3⋮vxvx+1vx+2⋮vx+yv1v2v3⋮vxvx+1vx+2…vx+y−y00…0xyxy…xy0−y0…0xyxy…xy00−y…0xyxy…xy⋮……⋱…………⋮000…−yxyxy…xyxyxyxy…xy−x0…0xyxyxy…xy0−x…0⋮……⋱………⋱⋮xyxyxy…xy00…−x.
Next, we have four cases to discuss.Case I (y=x+1 with x≥1): eigenvalues of LDPAKx,y are(13)−y+x2±124y3x3+11−time,−xy−1−times,−yx−1−times.The required spectrum can be obtained by these eigenvalues. Furthermore, by (3), we have(14)LEDPAKx,y=−y+x2+124y3x3+1+−y+x2−124y3x3+1+y−1.−x+x−1.−y=2xy.Case II (y≠x+1 with y>x≥1): we get the following eigenvalues of LDPAKx,y:(15)−y+x2±124x3y3+y−x21−time,−xy−1−times,−yx−1−times.The required spectrum can be obtained by these eigenvalues. Moreover, by (3), we have(16)LEDPAKx,y=−y+x2+124x3y3+y−x2+−y+x2−124x3y3+y−x2+y−1.−x+x−1.−y=2xy.Case III (x=y≥1): eigenvalues of LDPAKx,x are as follows:(17)xx2−1,1−time,−xx2+1,1−time,−x,2x−1−times.These eigenvalues provide the required spectrum. Furthermore, by (3), we have(18)LEDPAKx,x=xx2−1+−xx2+1+2x−1.−x=2x3+x2−x.Case IV (x=1 and y≥1): we get eigenvalues of LDPAK1,y as follows:(19)−1+y2±124y3+1−y2,1−time,−1,y−1−times.These eigenvalues provide the required spectrum. Using (3), we have the following energy of K1,y:(20)LEDPAK1,y=−1+y2+124y3+1−y2+−1+y2−124y3+1−y2+y−1.−1=2y.
It completes the proof.


### 2.3. Friendship GraphsFx

A friendship graphFx has 2x+1 vertices, and it can be assembled by connecting x clones of the cycle C3 with a common vertex. Let the vertex set of ith copy of C3 be v1i,v2i,v3i, where 1≤i≤x. Let the common vertex be v=v11=v12=…=v1x. Then, the vertex set of Fx is(21)v⋃i=1xv2i,v3i.Theorem 3.
Forx≥2, let a friendship graph Fx; then,(22)LSpDPAFx=2−61−x±32x3+1+x2x−1x1,LEDPAFx=10x−4.Proof.
InFx, dv=2x and dv2i=dv3i=2 for 1≤i≤x. Then, the Laplacian degree product adjacency matrix is as follows:(23)LDPAFx=vv21v31v22v32⋮v2xv3xvv21v31v22v32…v2xv3x−2x4x4x4x4x…4x4x4x−2400…004x4−200…004x00−24…004x004−2…00⋮⋮⋮⋮⋮⋱⋮⋮4x0000…−244x0000…4−2.
The eigenvalues of Laplacian degree product adjacency matrix ofFx are(24)2,x−1−times,−6,x−times,1−x±32x3+1+x2,1−time.
The required spectrum can be obtained by these eigenvalues. These eigenvalues provide the following energy:(25)LEDPAFx=x−1.2+x.−6+1−x+32x3+1+x2+1−x−32x3+1+x2=10x−4.


### 2.4. Corona Products of 3 and 4 Cycles with Null Graphs

The corona product of graphsΓ and Ω is expressed as Γ°Ω. It can be made by drawing one copy of Γ and ∣VΓ∣ copies of Ω and connecting the ith vertex of Γ with each vertex of ith copy of Ω [9–11]. Let Γ be an x-cycle Cx with vertices v1,v2,…,vx and Ω be a null graph Nk. Then, the vertex set of Cx°Nk is(26)VCx⋃j=1xvij;1≤i≤k,where the set vij;1≤i≤k is the vertex set of jth copy of Nk in Cx°Nk. In this portion, we evaluate the Laplacian degree product spectrum and energy of Cx°Nk for x=3 and 4.Theorem 4.
Fork≥1, let the corona product be Cx°Nk; then,(27)LSpDPAC3°Nk=122k2+7k+5±4k4+32k3+93k2+114k+49−12k2+5k+7∓k4+14k3+51k2+66k+25−1123k−1,LEDPAC3°Nk=4k2+5k+4.Proof.
Note thatdvj=k+2, for each j=1,2,3, and dvij=1, for each 1≤j≤3 and 1≤i≤k. For the convenience, we let k+2=α. Then, the Laplacian degree product adjacency matrix of Cx°Nk is(28)v1v2v3v11⋮vk1v12⋮vk2v13⋮vk3v1v2v3v11…vk1v12…vk2v13…vk3−αα2α2α…α0…00…0α2−αα20…0α…α0…0α2α2−α0…00…0α…αα00−1…00…00…0⋮⋮⋮⋮⋱⋮⋮…⋮⋮…⋮α000…−10…00…00α00…0−1…00…0⋮⋮⋮⋮…⋮⋮⋱⋮⋮…⋮0α00…00…−10…000α0…00…0−1…0⋮⋮⋮⋮…⋮⋮…⋮⋮⋱⋮00α0…00…00…−1
The eigenvalues obtained from the above matrix ofCx°Nk are(29)2k2+7k+52±4k4+32k3+93k2+114k+494,1−times,−k2+5k+72±12k4+14k3+51k2+66k+25,2−times,−1,3k−1−times.
Then, the required spectrum can be obtained by these eigenvalues. Also, by (3), we have(30)LEDPAC3°Nk=2k2+7k+52+4k4+32k3+93k2+114k+494+2k2+7k+52−4k4+32k3+93k2+114k+494+−k2+5k+72+12k4+14k3+51k2+66k+25+−k2+5k+72−12k4+14k3+51k2+66k+25+3k−1−1=4k2+5k+4.Theorem 5.
Fork≥1, let the corona product be Cx°Nk; then, LSpDPACx°Nk is(31)−122k2+9k+11∓4k4+40k3+133k2+178k+81−12k+3∓4k3+17k2+18k+1122k2+7k+5±4k4+32k3+93k2+114k+49−11214k−1LEDPAC4°Nk=4k2+22k+18.Proof.
Note thatdvj=k+2, for eachj=1,2,3,4, and dvij=1, for each 1≤j≤4 and 1≤i≤k. For the convenience, we let k+2=α. Then, the Laplacian degree product adjacency matrix of C4°Nk is as(32)v1v2v3v4v11⋮vk1v14⋮vk4v1v2v3v4v11…vk1v14…vk4−αα20α2α…α0…0α2−αα200…00…00α2−αα20…00…0α20α2−α0…0α…αα000−1…00…0⋮⋮⋮⋮⋮⋱⋮⋮…⋮α0000…−10…0000α0…0−1…0⋮⋮⋮⋮⋮…⋮⋮⋱⋮000α0…00…−1.
The eigenvalues of Laplacian degree product adjacency matrix ofC4°Nk are(33)−2k2+9k+112±4k4+40k3+133k2+178k+814,1−time,−k+32±4k3+17k2+18k+14,2−times,2k2+7k+52±4k4+32k3+93k2+114k+494,1−time,−14k−1−times.
Then, the required spectrum can be obtained by these eigenvalues. Furthermore, by (3), we have(34)LEDPAC4°Nk=−2k2+9k+112+4k4+40k3+133k2+178k+814+−2k2+9k+112−4k4+40k3+133k2+178k+814+2.−k+32+4k3+17k2+18k+14+2.−k+32−4k3+17k2+18k+14+2k2+7k+52+4k4+32k3+93k2+114k+494+2k2+7k+52−4k4+32k3+93k2+114k+494+4k−1.−1=22k2+11k+9.


## 2.1. Complete GraphsKx

Letv1,v2…,vx be the vertex set of Kx; then, the following result provides the Laplacian degree product adjacency spectrum and energy of Kx.Theorem 1.
Forx≥2, let Kx be a complete graph. Then,(6)LSpDPAKx=xx2−3x+2x1−x1x−1,and Laplacian degree product adjacency energy of Kx is 22x−3-times the size of Kx.Proof.
First of all note thatdvi=x−1, for each 1≤i≤x. Accordingly, we have the following Laplacian degree product adjacency matrix:(7)LDPAKx=v1v2v3…vxv1v2v3…vx1−x1−x21−x2…1−x21−x21−x1−x2…1−x21−x21−x21−x…1−x2⋮⋮⋮⋱⋮1−x21−x21−x2…1−x.
Eigenvalues ofLDPAKx are(8)x1−x,x−1−times,xx2−3x+2,1−time.
These eigenvalues provide the required spectrum. Moreover, by (3), we have(9)LEDPAKx=x−1x1−x+xx2−3x+2=22x−3x2.
Since the size ofKx is x2, so the result is proved.


## 2.2. Complete Bipartite GraphsKx,y

Let a complete bipartite graphKx,y with vertex sets VxΓ=v1,v2,…,vx and VyΓ=vx+1,vx+2,…,vy be as partitions. The order and the size of Kx,y graph are x+y and xy, respectively. Then, the Laplacian degree product adjacency spectrum and energy of Kx,y can be obtained from the following result.Theorem 2.
Forx,y≥1, a complete bipartite graph Kx,y, then(10)LSpDPAKx,y=−x+y2±124y3x3+1−x−y1y−1x−1,y=x+1withx≥1,−x+y2±124y3x3+y−x2−x−y1y−1x−1,y>x≥1withy≠x+1,xx2−1−xx2+1−x112x−1,x=y≥1,−1+y2±124y3+1−y2−11y−1x=1andy≥1.
Moreover,(11)LEDPAKx,y=2x3+x2−x,wheneverx=y≥1,2xy,otherwise.Proof.
Note thatdvi=y, for each 1≤i≤x and dvj=x, for each x+1≤j≤y. Accordingly, the Laplacian degree product adjacency matrix of Kx,y is(12)LDPAKx,y=v1v2v3⋮vxvx+1vx+2⋮vx+yv1v2v3⋮vxvx+1vx+2…vx+y−y00…0xyxy…xy0−y0…0xyxy…xy00−y…0xyxy…xy⋮……⋱…………⋮000…−yxyxy…xyxyxyxy…xy−x0…0xyxyxy…xy0−x…0⋮……⋱………⋱⋮xyxyxy…xy00…−x.
Next, we have four cases to discuss.Case I (y=x+1 with x≥1): eigenvalues of LDPAKx,y are(13)−y+x2±124y3x3+11−time,−xy−1−times,−yx−1−times.The required spectrum can be obtained by these eigenvalues. Furthermore, by (3), we have(14)LEDPAKx,y=−y+x2+124y3x3+1+−y+x2−124y3x3+1+y−1.−x+x−1.−y=2xy.Case II (y≠x+1 with y>x≥1): we get the following eigenvalues of LDPAKx,y:(15)−y+x2±124x3y3+y−x21−time,−xy−1−times,−yx−1−times.The required spectrum can be obtained by these eigenvalues. Moreover, by (3), we have(16)LEDPAKx,y=−y+x2+124x3y3+y−x2+−y+x2−124x3y3+y−x2+y−1.−x+x−1.−y=2xy.Case III (x=y≥1): eigenvalues of LDPAKx,x are as follows:(17)xx2−1,1−time,−xx2+1,1−time,−x,2x−1−times.These eigenvalues provide the required spectrum. Furthermore, by (3), we have(18)LEDPAKx,x=xx2−1+−xx2+1+2x−1.−x=2x3+x2−x.Case IV (x=1 and y≥1): we get eigenvalues of LDPAK1,y as follows:(19)−1+y2±124y3+1−y2,1−time,−1,y−1−times.These eigenvalues provide the required spectrum. Using (3), we have the following energy of K1,y:(20)LEDPAK1,y=−1+y2+124y3+1−y2+−1+y2−124y3+1−y2+y−1.−1=2y.
It completes the proof.


## 2.3. Friendship GraphsFx

A friendship graphFx has 2x+1 vertices, and it can be assembled by connecting x clones of the cycle C3 with a common vertex. Let the vertex set of ith copy of C3 be v1i,v2i,v3i, where 1≤i≤x. Let the common vertex be v=v11=v12=…=v1x. Then, the vertex set of Fx is(21)v⋃i=1xv2i,v3i.Theorem 3.
Forx≥2, let a friendship graph Fx; then,(22)LSpDPAFx=2−61−x±32x3+1+x2x−1x1,LEDPAFx=10x−4.Proof.
InFx, dv=2x and dv2i=dv3i=2 for 1≤i≤x. Then, the Laplacian degree product adjacency matrix is as follows:(23)LDPAFx=vv21v31v22v32⋮v2xv3xvv21v31v22v32…v2xv3x−2x4x4x4x4x…4x4x4x−2400…004x4−200…004x00−24…004x004−2…00⋮⋮⋮⋮⋮⋱⋮⋮4x0000…−244x0000…4−2.
The eigenvalues of Laplacian degree product adjacency matrix ofFx are(24)2,x−1−times,−6,x−times,1−x±32x3+1+x2,1−time.
The required spectrum can be obtained by these eigenvalues. These eigenvalues provide the following energy:(25)LEDPAFx=x−1.2+x.−6+1−x+32x3+1+x2+1−x−32x3+1+x2=10x−4.


## 2.4. Corona Products of 3 and 4 Cycles with Null Graphs

The corona product of graphsΓ and Ω is expressed as Γ°Ω. It can be made by drawing one copy of Γ and ∣VΓ∣ copies of Ω and connecting the ith vertex of Γ with each vertex of ith copy of Ω [9–11]. Let Γ be an x-cycle Cx with vertices v1,v2,…,vx and Ω be a null graph Nk. Then, the vertex set of Cx°Nk is(26)VCx⋃j=1xvij;1≤i≤k,where the set vij;1≤i≤k is the vertex set of jth copy of Nk in Cx°Nk. In this portion, we evaluate the Laplacian degree product spectrum and energy of Cx°Nk for x=3 and 4.Theorem 4.
Fork≥1, let the corona product be Cx°Nk; then,(27)LSpDPAC3°Nk=122k2+7k+5±4k4+32k3+93k2+114k+49−12k2+5k+7∓k4+14k3+51k2+66k+25−1123k−1,LEDPAC3°Nk=4k2+5k+4.Proof.
Note thatdvj=k+2, for each j=1,2,3, and dvij=1, for each 1≤j≤3 and 1≤i≤k. For the convenience, we let k+2=α. Then, the Laplacian degree product adjacency matrix of Cx°Nk is(28)v1v2v3v11⋮vk1v12⋮vk2v13⋮vk3v1v2v3v11…vk1v12…vk2v13…vk3−αα2α2α…α0…00…0α2−αα20…0α…α0…0α2α2−α0…00…0α…αα00−1…00…00…0⋮⋮⋮⋮⋱⋮⋮…⋮⋮…⋮α000…−10…00…00α00…0−1…00…0⋮⋮⋮⋮…⋮⋮⋱⋮⋮…⋮0α00…00…−10…000α0…00…0−1…0⋮⋮⋮⋮…⋮⋮…⋮⋮⋱⋮00α0…00…00…−1
The eigenvalues obtained from the above matrix ofCx°Nk are(29)2k2+7k+52±4k4+32k3+93k2+114k+494,1−times,−k2+5k+72±12k4+14k3+51k2+66k+25,2−times,−1,3k−1−times.
Then, the required spectrum can be obtained by these eigenvalues. Also, by (3), we have(30)LEDPAC3°Nk=2k2+7k+52+4k4+32k3+93k2+114k+494+2k2+7k+52−4k4+32k3+93k2+114k+494+−k2+5k+72+12k4+14k3+51k2+66k+25+−k2+5k+72−12k4+14k3+51k2+66k+25+3k−1−1=4k2+5k+4.Theorem 5.
Fork≥1, let the corona product be Cx°Nk; then, LSpDPACx°Nk is(31)−122k2+9k+11∓4k4+40k3+133k2+178k+81−12k+3∓4k3+17k2+18k+1122k2+7k+5±4k4+32k3+93k2+114k+49−11214k−1LEDPAC4°Nk=4k2+22k+18.Proof.
Note thatdvj=k+2, for eachj=1,2,3,4, and dvij=1, for each 1≤j≤4 and 1≤i≤k. For the convenience, we let k+2=α. Then, the Laplacian degree product adjacency matrix of C4°Nk is as(32)v1v2v3v4v11⋮vk1v14⋮vk4v1v2v3v4v11…vk1v14…vk4−αα20α2α…α0…0α2−αα200…00…00α2−αα20…00…0α20α2−α0…0α…αα000−1…00…0⋮⋮⋮⋮⋮⋱⋮⋮…⋮α0000…−10…0000α0…0−1…0⋮⋮⋮⋮⋮…⋮⋮⋱⋮000α0…00…−1.
The eigenvalues of Laplacian degree product adjacency matrix ofC4°Nk are(33)−2k2+9k+112±4k4+40k3+133k2+178k+814,1−time,−k+32±4k3+17k2+18k+14,2−times,2k2+7k+52±4k4+32k3+93k2+114k+494,1−time,−14k−1−times.
Then, the required spectrum can be obtained by these eigenvalues. Furthermore, by (3), we have(34)LEDPAC4°Nk=−2k2+9k+112+4k4+40k3+133k2+178k+814+−2k2+9k+112−4k4+40k3+133k2+178k+814+2.−k+32+4k3+17k2+18k+14+2.−k+32−4k3+17k2+18k+14+2k2+7k+52+4k4+32k3+93k2+114k+494+2k2+7k+52−4k4+32k3+93k2+114k+494+4k−1.−1=22k2+11k+9.


## 3. Appendix

In [6], Mirajkar and Doddamani considered the corona product C4°Nk−2 for k≥3 (also called thorny cycle rings C4,k) and investigated its energy and spectrum on the base of degree product adjacency matrix. During computations of our results on C4°Nk−2, the eigenvalues investigated in [6] were found incorrect. In this section, we provide the correct energy and spectrum of C4°Nk−2. First of all, note that the degree product adjacency matrix of C4°Nk−2 is(35)DPAC4°Nk−2=v1v2v3v4v11v21⋮v14v24v1v2v3v4v11v21…v14v240k20k2kk…00k20k2000…000k20k200…00k20k2000…kkk00000…00k00000…00⋮⋮⋮⋮⋮⋮⋱⋮⋮000k00…00000k00…00Mirajkar and Doddamani’s investigated eigenvalues are(36)kk2+2±k,1−time,−kk2+2±k,1−time,±k2,2−times,0,4k−3−times.Whereas, the correct eigenvalues ofDPAC4°Nk−2 are(37)±k2+kk2+k−2,1−time,∓k2−kk2+k−2,1−time,±kk−2,2−times,0,4k−3−times.Accordingly, we get the following energy and spectrum ofC4°Nk−2 in the corrected form:(38)EDPAC4°Nk−2=k2+kk2+k−2+−k2+kk2+k−2+−k2−kk2+k−2+k2−kk2+k−2+2.−kk−2+2.kk−2,SpDPAC4°Nk−2=±k2+kk2+k−2∓k2−kk2+k−2±kk−201124k−3.


## 4. Accomplishment Remarks

In this study, we construct the general formulas for spectrums and energies of four different families of graphs, by using Laplacian degree product adjacency matrix. We also obtained faultless and correct eigenvalues ofC4°Nk−2, which were defined in [6].



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