Signless Laplacian Polynomial and Characteristic Polynomial of a Graph

Abstract

The signless Laplacian polynomial of a graph 𝐺 is the characteristic polynomial of the matrix 𝑄(𝐺)=𝐷(𝐺)+𝐴(𝐺), where 𝐷(𝐺) is the diagonal degree matrix and 𝐴(𝐺) is the adjacency matrix of 𝐺. In this paper we express the signless Laplacian polynomial in terms of the characteristic polynomial of the induced subgraphs, and, for regular graph, the signless Laplacian polynomial is expressed in terms of the derivatives of the characteristic polynomial. Using this we obtain the characteristic polynomial of line graph and subdivision graph in terms of the characteristic polynomial of induced subgraphs.

1. Introduction

Let 𝐺 be a simple graph with 𝑛 vertices and 𝑚 edges. Let the vertex set of 𝐺 be 𝑉(𝐺)={𝑣₁,𝑣₂,…,𝑣_𝑛}. Let deg_𝐺(𝑣) denote the degree of a vertex 𝑣 in 𝐺. The adjacency matrix of a graph 𝐺 is 𝐴(𝐺)=[𝑎_𝑖𝑗], where 𝑎_𝑖𝑗=1 if 𝑣_𝑖 is adjacent to 𝑣_𝑗 and 𝑎_𝑖𝑗=0, otherwise. The characteristic polynomial of a graph 𝐺 is defined as 𝜙(𝐺∶𝜆)=det(𝜆𝐼−𝐴(𝐺)), where 𝐼 is an identity matrix of order 𝑛. The degree matrix of a graph 𝐺 is the diagonal matrix 𝐷(𝐺)=diag[𝑑₁,𝑑₂,…,𝑑_𝑛], where 𝑑_𝑖=deg_𝐺(𝑣_𝑖), 𝑖=1,2,…,𝑛. The matrix 𝐶(𝐺)=𝐷(𝐺)−𝐴(𝐺), is called the Laplacian matrix and the matrix 𝑄(𝐺)=𝐷(𝐺)+𝐴(𝐺) is called the signless Laplacian matrix or 𝑄-matrix of 𝐺. The characteristic polynomial of 𝐶(𝐺), defined as 𝐶(𝐺∶𝜆)=det(𝜆𝐼−𝐶(𝐺)), is called the Laplacian polynomial of 𝐺. The characteristic polynomial of 𝑄(𝐺), defined as 𝑄(𝐺∶𝜆)=det(𝜆𝐼−𝑄(𝐺)), is called the signless Laplacian polynomial or 𝑄-polynomial of a graph 𝐺, where 𝐼 is an identity matrix of order 𝑛.

Several results on Laplacian of 𝐺 are reported in the literature (see, e.g., [1–4]). Recently signless Laplacian attracted the attention of researchers [5–12]. In [13], the Laplacian polynomial of a graph is expressed in terms of the characteristic polynomial of the induced subgraphs. In this paper we express signless Laplacian polynomial of a graph in terms of the characteristic polynomial of its induced subgraphs. Further the signless Laplacian polynomial of a regular graph is expressed in terms of the derivatives of its characteristic polynomial. Using these results, we express characteristic polynomial of line graph and of subdivision graph in terms of the characteristic polynomial of its induced subgraphs.

We use standard terminology of graph theory [14].

2. Signless Laplacian Polynomial in terms of Characteristic Polynomial

Let the set 𝑀_𝑘(𝐺)={𝑆∣𝑆⊆𝑉(𝐺)and|𝑆|=𝑘}, 𝑘=0,1,2,…,𝑛. Note that |𝑀_𝑘(𝐺)|=(^𝑛_𝑘). We denote the product of degrees of the vertices of 𝐺 which belongs to 𝑆 by 𝑃_𝐺(𝑆), that is, 𝑃_𝐺(𝑆)=∏_𝑣∈𝑆deg_𝐺(𝑣). The graph 𝐺−𝑆 is an induced subgraph of 𝐺 with vertex set 𝑉(𝐺)−𝑆. If 𝑆=𝑉(𝐺), then 𝐺−𝑆=𝐾₀, a graph without vertices. Note that 𝜙(𝐾₀∶𝜆)=1.

Theorem 1 (see [15] (𝑘th derivative)). Let 𝐺 be a graph with 𝑛 vertices, then 𝑑^𝑘𝑑𝜆^𝑘𝜙(𝐺∶𝜆)=𝑘!_{𝑆∈𝑀𝑘(𝐺)}𝜙(𝐺−𝑆∶𝜆),0≤𝑘≤𝑛. (1)

Theorem 2. Let 𝐺 be a graph with 𝑛 vertices, then 𝑄(𝐺∶𝜆)=^𝑛_𝑘=0(−1)^𝑘_{𝑆∈𝑀𝑘(𝐺)}𝑃_𝐺(𝑆)𝜙(𝐺−𝑆∶𝜆). (2)

Proof. Let 𝐴(𝐺)=[𝑎_𝑖𝑗], 1≤𝑖,𝑗≤𝑛 be the adjacency matrix of 𝐺 and 𝐷(𝐺)=diag[𝑑₁,𝑑₂,…,𝑑_𝑛], where 𝑑_𝑖=deg_𝐺(𝑣_𝑖), 𝑖=1,2,…,𝑛. Then, 𝑄()((𝐺((𝐺)(𝐺=𝐺∶𝜆=det𝜆𝐼−𝑄))=det𝜆𝐼−𝐷−𝐴))𝜆−𝑑₁−𝑎₁₂−𝑎₁₃⋯−𝑎_1𝑛−𝑎₂₁𝜆−𝑑₂−𝑎₂₃⋯−𝑎_2𝑛−𝑎₃₁−𝑎₃₂𝜆−𝑑₃⋯−𝑎_3𝑛⋮⋮−𝑎_𝑛1−𝑎_𝑛2−𝑎_𝑛3⋯𝜆−𝑑_𝑛. (3)
Splitting the determinant of (3) as a sum of two determinants, we get 𝑄(𝐺∶𝜆)=𝜆−𝑎₁₂−𝑎₁₃⋯−𝑎_1𝑛−𝑎₂₁𝜆−𝑑₂−𝑎₂₃⋯−𝑎_2𝑛−𝑎₃₁−𝑎₃₂𝜆−𝑑₃⋯−𝑎_3𝑛⋮⋮−𝑎_𝑛1−𝑎_𝑛2−𝑎_𝑛3⋯𝜆−𝑑_𝑛+−𝑑₁00⋯0−𝑎₂₁𝜆−𝑑₂−𝑎₂₃⋯−𝑎_2𝑛−𝑎₃₁−𝑎₃₂𝜆−𝑑₃⋯−𝑎_3𝑛⋮⋮−𝑎_𝑛1−𝑎_𝑛2−𝑎_𝑛3⋯𝜆−𝑑_𝑛. (4)
Again splitting each of the determinants of (4) as a sum of two determinants and continuing the procedure in succession, at 𝑛th step we get 𝑄()=|(𝐺+𝐺∶𝜆𝜆𝐼−𝐴)|_{1≤𝑖≤𝑛}−𝑑_𝑖𝜆𝐼−𝐴𝐺−𝑣_𝑖+_{1≤𝑖<𝑗≤𝑛}−𝑑_𝑖−𝑑_𝑗𝜆𝐼−𝐴𝐺−𝑣_𝑖−𝑣_𝑗+⋯+_{1≤𝑖<𝑗<⋯<𝑙≤𝑛}−𝑑_𝑖−𝑑_𝑗⋯−𝑑_𝑙×𝜆𝐼−𝐴𝐺−𝑣_𝑖−𝑣_𝑗−⋯−𝑣_𝑙. (5)
(in the above expression, 𝐼 stands for the identity matrix of the appropriate order and 𝐺−𝑣_𝑖 is a vertex deleted subgraph of 𝐺) and𝑄()=|(𝐺+()𝐺∶𝜆𝜆𝐼−𝐴)|−1_{𝑆∈𝑀1(𝐺)}𝑃_𝐺(𝑆)|(+()𝜆𝐼−𝐴𝐺−𝑆)|−1²_{𝑆∈𝑀2(𝐺)}𝑃_𝐺(𝑆)|(+()𝜆𝐼−𝐴𝐺−𝑆)|+⋯−1^𝑛_{𝑆∈𝑀𝑛(𝐺)}𝑃_𝐺(𝑆(=)|𝜆𝐼−𝐴𝐺−𝑆)|_𝑛𝑘=0(−1)^𝑘_{𝑆∈𝑀𝑘(𝐺)}𝑃_𝐺(𝑆)|(=𝜆𝐼−𝐴𝐺−𝑆)|_𝑛𝑘=0(−1)^𝑘_{𝑆∈𝑀𝑘(𝐺)}𝑃_𝐺(𝑆)𝜙(𝐺−𝑆∶𝜆). (6)

Corollary 3. If 𝐺 is an 𝑟-regular graph with 𝑛 vertices, then 𝑄(𝐺∶𝜆)=^𝑛_𝑘=0(−1)^𝑘𝑟^𝑘𝑑𝑘!𝑑𝜆^𝑘𝑘𝜙(𝐺∶𝜆). (7)

Proof. The graph 𝐺 is an 𝑟-regular graph, therefore 𝑃_𝐺(𝑆)=𝑟^𝑘 for all 𝑆∈𝑀_𝑘(𝐺). Substituting this in (2) gives that 𝑄(𝐺∶𝜆)=^𝑛_𝑘=0(−1)^𝑘_{𝑆∈𝑀𝑘(𝐺)}𝑟^𝑘𝜙()=𝐺−𝑆∶𝜆_𝑛𝑘=0(−1)^𝑘𝑟^𝑘_{𝑆∈𝑀𝑘(𝐺)}𝜙()=𝐺−𝑆∶𝜆_𝑛𝑘=0(−1)^𝑘𝑟^𝑘𝑑𝑘!𝑑𝜆^𝑘𝑘𝜙(𝐺∶𝜆)from(1). (8)

3. Characteristic Polynomial of Line Graph and Subdivision Graph

The line graph 𝐿(𝐺) of a graph 𝐺 is the graph whose vertex set has one-to-one correspondence to the edge set of 𝐺 and two vertices in 𝐿(𝐺) are adjacent if and only if the corresponding edges have a vertex in common in 𝐺.

Theorem 4. Let 𝐺 be a graph with 𝑛 vertices and 𝑚 edges and 𝐿(𝐺) is the line graph of 𝐺, then 𝜙(𝐿(𝐺))=()∶𝜆𝜆+2^𝑚−𝑛×^𝑛_𝑘=0(−1)^𝑘_{𝑆∈𝑀𝑘(𝐺)}𝑃_𝐺(𝑆)𝜙(𝐺−𝑆∶𝜆+2). (9)

Proof. If 𝐺 is a graph with 𝑛 vertices and 𝑚 edges, then [8] 𝜙(𝐿(𝐺)∶𝜆)=(𝜆+2)^𝑚−𝑛𝑄(𝐺∶𝜆+2). (10)
From (2), 𝑄()=𝐺∶𝜆+2_𝑛𝑘=0(−1)^𝑘_{𝑆∈𝑀𝑘(𝐺)}𝑃_𝐺(𝑆)𝜙(𝐺−𝑆∶𝜆+2). (11)
Substituting (11) in (10), the result follows.

Corollary 5. If 𝐺 is an 𝑟-regular graph with 𝑛 vertices, then 𝜙(𝐿(𝐺))=()∶𝜆𝜆+2^{𝑛𝑛(𝑟−2)/2}_𝑘=0(−1)^𝑘𝑟^𝑘𝑑𝑘!𝑑𝑥^𝑘𝑘𝜙(𝐺∶𝑥)|_𝑥=𝜆+2. (12)

Proof. The graph 𝐺 is an 𝑟-regular graph, therefore 𝑃_𝐺(𝑆)=𝑟^𝑘 for all 𝑆∈𝑀_𝑘(𝐺) and the number of edges of 𝐺 is 𝑚=𝑛𝑟/2.
Therefore, from (9), 𝜙(𝐿(𝐺))=()∶𝜆𝜆+2^{𝑛𝑛(𝑟−2)/2}_𝑘=0(−1)^𝑘_{𝑆∈𝑀𝑘(𝐺)}𝑟^𝑘𝜙()=()𝐺−𝑆∶𝜆+2𝜆+2^{𝑛𝑛(𝑟−2)/2}_𝑘=0(−1)^𝑘𝑟^𝑘_{𝑆∈𝑀𝑘(𝐺)}𝜙(𝐺−𝑆∶𝑥)_𝑥=𝜆+2=(𝜆+2)^{𝑛𝑛(𝑟−2)/2}_𝑘=0(−1)^𝑘𝑟^𝑘×𝑑𝑘!_𝑘𝑑𝑥^𝑘𝜙(𝐺∶𝑥)|_𝑥=𝜆+2from(1). (13)

The subdivision graph 𝑆(𝐺) of a graph 𝐺 is the graph obtained from 𝐺 by inserting a new vertex into every edge of 𝐺.

Theorem 6. Let 𝐺 be a graph with 𝑛 vertices, 𝑚 edges and 𝑆(𝐺) is the subdivision graph of 𝐺, then 𝜙(𝑆(𝐺)∶𝜆)=𝜆^{𝑛𝑚−𝑛}_𝑘=0(−1)^𝑘_{𝑆∈𝑀𝑘(𝐺)}𝑃_𝐺(𝑆)𝜙𝐺−𝑆∶𝜆². (14)

Proof. If 𝐺 is a graph with 𝑛 vertices and 𝑚 edges, then [8] 𝜙(𝑆(𝐺)∶𝜆)=𝜆^𝑚−𝑛𝑄𝐺∶𝜆². (15)
From (2), 𝑄𝐺∶𝜆²=^𝑛_𝑘=0(−1)^𝑘_{𝑆∈𝑀𝑘(𝐺)}𝑃_𝐺(𝑆)𝜙𝐺−𝑆∶𝜆². (16)
Substituting (16) in (15), the result follows.

Corollary 7. If 𝐺 is an 𝑟-regular graph with 𝑛 vertices, then 𝜙(𝑆(𝐺)∶𝜆)=𝜆^{𝑛𝑛(𝑟−2)/2}_𝑘=0(−1)^𝑘𝑟^𝑘𝑑𝑘!𝑑𝑥^𝑘𝑘𝜙(𝐺∶𝑥)|_𝑥=𝜆². (17)

Proof. As 𝐺 is an 𝑟-regular graph, 𝑃_𝐺(𝑆)=𝑟^𝑘 for all 𝑆∈𝑀_𝑘(𝐺) and the number of edges of 𝐺 is 𝑚=𝑛𝑟/2.
Therefore, from (14), 𝜙(𝑆(𝐺)∶𝜆)=𝜆^{𝑛𝑛(𝑟−2)/2}_𝑘=0(−1)^𝑘_{𝑆∈𝑀𝑘(𝐺)}𝑟^𝑘𝜙𝐺−𝑆∶𝜆²=𝜆^{𝑛𝑛(𝑟−2)/2}_𝑘=0(−1)^𝑘𝑟^𝑘_{𝑆∈𝑀𝑘(𝐺)}𝜙(𝐺−𝑆∶𝑥)_𝑥=𝜆2=𝜆^{𝑛𝑛(𝑟−2)/2}_𝑘=0(−1)^𝑘𝑟^𝑘𝑑𝑘!𝑑𝑥^𝑘𝑘𝜙(𝐺∶𝑥)|_𝑥=𝜆²from(1). (18)

4. Number of Spanning Trees

If 𝐺 is an 𝑟-regular graph, then 𝑄()=|(𝐺=|(𝐺)(𝐺=)(𝐺()𝐺∶𝜆𝜆𝐼−𝑄)|𝜆𝐼−𝐷−𝐴)||(𝜆−𝑟𝐼−𝐴)|since𝐷=𝑟𝐼=𝜙𝐺∶𝜆−𝑟 (19)

which gives that 𝑄(𝐺∶2𝑟−𝜆)=𝜙(𝐺∶𝑟−𝜆). (20)

Also for an 𝑟-regular graph 𝐺 [3, 16], 𝐶(𝐺∶𝜆)=(−1)^𝑛𝜙(𝐺∶𝑟−𝜆). (21)

Let 𝜏(𝐺) be the number of spanning trees of 𝐺, then [3, 16] 𝜏(𝐺)=(−1)^𝑛−1𝑛𝑑𝐶(𝑑𝜆𝐺∶𝜆)|_𝜆=0. (22)

Here we give the number of spanning trees using signless Laplacian.

Theorem 8. If 𝐺 is an 𝑟-regular graph with 𝑛 vertices, then 𝜏(𝐺)=1𝑛𝑑𝑄(𝑑𝜆𝐺∶𝜆)|_𝜆=2𝑟. (23)

Proof. From (22), 𝜏(𝐺)=(−1)^𝑛−1𝑛𝑑𝐶(𝑑𝜆𝐺∶𝜆)|_𝜆=0=(−1)^𝑛−1𝑛𝑑()𝑑𝜆−1^𝑛𝜙(𝐺∶𝑟−𝜆)|_𝜆=0(),=()from21−1^𝑛−1𝑛𝑑()𝑑𝜆−1^𝑛𝑄(𝐺∶2𝑟−𝜆)|_𝜆=0()=()from20−1^𝑛−1𝑛𝑑()𝑑𝜆−1^𝑛(−1)𝑄(𝐺∶𝜆)|_𝜆=2𝑟=1𝑛𝑑𝑄(𝑑𝜆𝐺∶𝜆)|_𝜆=2𝑟. (24)

5. Conclusion

Equation (2) establishes the relationship between the signless Laplacian polynomial and characteristic polynomial of a graph. Equations (9) and (14) express the characteristic polynomial of line graph and of subdivision graph in terms of the characteristic polynomial of a graph. As corollaries of these results, (12) and (17) give the characteristic polynomial of line graph and the subdivision graph of a regular graph in terms of the derivatives of the characteristic polynomial of a graph. These results are different from those obtained in [17, 18].