SPECTRA OF CLOSENESS LAPLACIAN AND CLOSENESS SIGNLESS LAPLACIAN OF GRAPHS

. For a graph 𝐺 with vertex set 𝑉 ( 𝐺 ) and 𝑢, 𝑣 ∈ 𝑉 ( 𝐺 ), the distance between vertices 𝑢 and 𝑣 in 𝐺 , denoted by 𝑑 𝐺 ( 𝑢, 𝑣 ), is the length of a shortest path connecting them and it is ∞ if there is no such a path, and the closeness of vertex 𝑢 in 𝐺 is 𝑐 𝐺 ( 𝑢 ) = ∑︀ 𝑤 ∈ 𝑉 ( 𝐺 ) 2 − 𝑑 𝐺 ( 𝑢,𝑤 ) . Given a graph 𝐺 that is not necessarily connected, for 𝑢, 𝑣 ∈ 𝑉 ( 𝐺 ), the closeness matrix of 𝐺 is the matrix whose ( 𝑢, 𝑣 )-entry is equal to 2 − 𝑑 𝐺 ( 𝑢,𝑣 ) if 𝑢 ̸ = 𝑣 and 0 otherwise, the closeness Laplacian is the matrix whose ( 𝑢, 𝑣 )-entry is equal to {︃ − 2 − 𝑑 𝐺 ( 𝑢,𝑣 ) if 𝑢 ̸ = 𝑣, 𝑐 𝐺 ( 𝑢 ) otherwise and the closeness signless Laplacian is the matrix whose ( 𝑢, 𝑣 )-entry is equal to {︃ 2 − 𝑑 𝐺 ( 𝑢,𝑣 ) if 𝑢 ̸ = 𝑣, 𝑐 𝐺 ( 𝑢 ) otherwise . We establish relations connecting the spectral properties of closeness Laplacian and closeness signless Laplacian and the structural properties of graphs. We give tight upper bounds for all nontrivial closeness Laplacian eigenvalues and characterize the extremal graphs, and determine all trees and unicyclic graphs that maximize the second smallest closeness Laplacian eigenvalue. Also, we give tight upper bounds for the closeness signless Laplacian eigenvalues and determine the trees whose largest closeness signless Laplacian eigenvalues achieve the first two largest values.


Introduction
We consider simple and undirected graphs.Let  be a graph with vertex set  () and edge set ().For ,  ∈  (), the distance between  and  in , denoted by   (, ), is the length of a shortest path from  to  in .Particularly,   (, ) = 0 for any  and   (, ) = ∞ if there is no path from  to  in .
For a graph  that is not necessarily connected, the closeness matrix of  is defined as [1] () = (  (, )) ,∈ () , where A general version has been considered, which is called the exponential distance matrix in [2] and the -distance matrix in [3], where, for any real number  ∈ (0, 1), the (, )-entry of the general version is    (,) if  ̸ =  and 0 otherwise.The closeness matrix of a graph deserves investigation because it combines some merits of the adjacency matrix and the distance matrix.It behalves as the adjacency matrix as both spectra are the union of the spectra of the corresponding matrices of the components of a graph.Moreover, it contains information of distances between different vertices.Note that distance matrix means only for connected graphs.Another motivation to consider the closeness matrix of a graph is the work of Dangalchev, who introduced in [4] the closeness of a graph , defined as () = ∑︀ ∈ ()   () with   () = ∑︀ ∈ ()∖{} 2 −  (,) .Dangalchev [5] Closeness is a measure of centrality, an important feature of communication and social networks.Rupnik Poklukar and Žerovnik [6] discussed the connection between the closeness of a graph and the early studied Hosoya polynomial (see also [7,8]), and they determined the graphs that minimize and maximize the closeness among several classes of graphs including trees and cacti.It is evident that for a graph , () is a symmetric nonnegative matrix.Moreover, () is irreducible if and only if  is connected.As () is symmetric, its eigenvalues are all real.We call them the closeness eigenvalues of .The largest closeness eigenvalue of a graph is called the spectral closeness used as a measure for networks [1], and the extremal values (minimum and maximum values) and the extremal graphs of spectral closeness have been determined there over various classes of graphs.Properties of other closeness eigenvalues, especially the second largest and the smallest closeness eigenvalues were explored in [9].Some results on the spectral properties of exponential distance matrix have been obtained in [10,11].
For  ∈ { L  (),  Q  ()}, we have by the well known Geršgorin discs theorem (Thm.6.1.1 in [13]) that | −   ()| ≤   () and so  ≥ 0 for some  ∈  ().That is, both L() and Q() are positive semi-definite.Denote by 1  the -dimensional column vector of all ones.Then L()1  = 0, so  L  () = 0.If  ≥ 2 and  is connected, then each entry of L() is not zero, so the matrix  obtained from L() by the deletion of, say, the last row and the last column is strictly diagonally dominant, implying that zero is not an eigenvalue of , from which it follows that the multiplicity of  L  () = 0 is one by the interlacing theorem (Thm.4.3.17 in [13]).So the number of 0 as a closeness Laplacian eigenvalue of a graph is equal to the number of components of the graph.Consequently, a graph  on  ≥ 2 vertices is connected if and only if  L −1 () > 0. This fact shows that the second smallest closeness Laplacian eigenvalue may be viewed as a distance-based 'algebraic connectivity' of a graph.Any closeness Laplacian eigenvalue that is not equal to zero is called a nontrivial one.
On the other hand, let x be a unit eigenvector of Q() associated to equivalently,  has at least a component with one or two vertices.Thus, if  is a connected graph with  ≥ 3 vertices, then  Q  () > 0. We establish some connections between the closeness Laplacian eigenvalues (closeness signless Laplacian eigenvalues, respectively) and the structural properties of graphs.On one hand, we give tight upper bounds for all nontrivial closeness Laplacian eigenvalues and characterize the extremal graphs and determine all trees and unicyclic graphs that maximize the second smallest closeness Laplacian eigenvalue.On the other hand, we give tight upper bounds for the closeness signless Laplacian eigenvalues and determine the trees whose largest closeness signless Laplacian eigenvalues achieve the first two largest values.

Preliminaries
For  ⊂  (), let  −  denote the graph obtained by removing each vertex of  (and all associated incident edges), and we write  −  for  − {} for  ∈  ().For  ⊆ (),  −  denotes the graph obtained from  by removing all edges of , and we write  −  for  − {} for  ∈ ().Denote by  the complement of a graph .For a set  ⊆ (),  +  denotes the graph obtained from  by adding all elements of  as edges, and we write  +  for  + {} for  / ∈ ().For vertex disjoint graphs  1 and  2 , let  1 ∪  2 be the (vertex disjoint) union of  1 and  2 , and Let   and   be the -vertex complete graph and path, respectively.Let  1,...,  be the complete -partite graph with   vertices in the th partite set for  = 1, . . ., , where  ≥ 2 and   ≥ 1.For positive integers  and  with 1 ≤  ≤ −2 2 , let  , be the tree on  vertices obtained from a path on two vertices by attaching  and  −  − 2 pendant vertices to its end vertices, respectively.
The degree of a vertex  in a graph  is the number of vertices that are adjacent to  in , denoted by   ().A vertex  is called a pendant vertex if   () = 1.For a graph  with  ∈  () and  ̸ ∈  (), we say that the graph  with  () =  () ∪ {} and () = () ∪ {} is obtained from  by attaching a pendant vertex at .
Given a graph , denote by Deg() the vertex degree diagonal matrix of a graph .The adjacency matrix of  is the matrix () = (  ) ,∈ () with   = 1 if  and  are adjacent and 0 otherwise.Then the Laplacian of  is the matrix () = Deg() − () and the signless Laplacian of  is () = Deg() + ().Both Laplacians have been extensively studied [14].
For a graph  with  () = { 1 , . . .,   }, a vector x = ( 1 , . . .,   ) ⊤ can be viewed as a function defined on  () that maps   to   .In this case,   is said to be the entry of x at  ∈  ().

Closeness Laplacian eigenvalues
Firstly, we recall some facts on the Laplacian eigenvalues of a graph.Let  be a connected graph on  vertices.
Proof.By Proposition 2.1, we have which, together with Proposition 3.1, implies that the closeness Laplacian spectrum of  is , 0 }︂ .
This completes the proof.
Lemma 3.3.Let  be a connected graph on  ≥ 2 vertices.Then there is a nonzero closeness Laplacian eigenvalue with multiplicity  − 1 if and only if  ∼ =   .
Proof.If  ∼ =   , then it is obvious that  2 is a nonzero closeness Laplacian eigenvalue with multiplicity  − 1. Suppose that  is a nonzero closeness Laplacian eigenvalue with multiplicity −1.Then L() has eigenvalues  with multiplicity  − 1 and 0 with multiplicity one.So for some  ×  orthonormal matrix  ,  ⊤ L() is a diagonal matrix with (, )-entry to be  for  = 1, . . .,  − 1 and 0 for  = .Let x = ( 1 , . . .,   ) be the last row vector of That is, L() =   − x ⊤ x.For  = 1, . . ., , considering the sum of entries of the th row of L(), we have implying that  1 = . . .=   := .This forces that the entries of L() outside the main diagonal are all equal, implying that  ∼ =   .
By Theorem 3.2 and Lemma 3.3, we immediately have the following consequence.
Corollary 3.4.Let  be a connected graph on  ≥ 2 vertices.Then with equality if and only if  ∼ =   .
Rupnik Poklukar and Žerovnik [6] noted that if  is a tree on  ≥ 2 vertices, then () ≤ ( 1,−1 ) = with equality if and only if  ∼ =  1,−1 .For a graph  with  ∈  (), we denote by   () the neighborhood of  in  (that is, the set of vertices that are adjacent to  in ).The following is Corollary 3.1 in [6].Let  be a nontrivial connected graph.Let  and  be two vertices of .Let  , (, ) be the graph obtained from  by attaching  pendant vertices at  and  pendant vertices at , where ,  ≥ 0, see Figure 2. Particularly,  , (0, 0) = .
The following is Corollary 3. Suppose that  = 3. ).By an easy direct calculation, one has , as desired.
Theorem 3.8.Let  be a tree on  ≥ 3 vertices.Then with equality if and only if Proof.The result is trivial if  = 3, and it follows easily if  = 4 as as desired.
Denote by  , with 3 ≤  ≤  the unicyclic graph on  vertices obtained from the cycle   by attaching  −  pendant vertices at a vertex.Denote by  * , with 3 ≤  ≤  − 2 the unicyclic graph on  vertices obtained from  +1, by attaching  −  − 1 pendant vertices at the pendant vertex, see Figure 3.
Let  1  be the unicyclic graph with  ≥ 5 vertices obtained from  −1,3 by attaching a pendant vertex at a vertex of degree two.Let  2  be the unicyclic graph with  ≥ 7 vertices obtained from  1 −1 by attaching a pendant vertex at the vertex of degree three.Let  +  be the unicyclic graph with  ≥ 5 vertices obtained from  −1,3 by attaching a pendant vertex at the vertex of degree one, see Figure 4. Lemma 3.9.Among unicyclic graphs on  vertices with girth three,  ,3 with  ≥ 4,  , respectively.
Proof.Let () be the set of unicyclic graphs on  vertices with girth three.The fact that  ,3 is the only graph in () that has the largest closeness follows from Lemmas 3.5 and 3.6, and by a direct calculation, ( ,3 ) =  2 + 4 .Let  ∈ () ∖ { ,3 }.By Lemmas 3.5 and 3.6, the maximum values of () is achieved only by one of  1  ,  +  and  * ,3 .By a direct calculation, we have and So it is evident that for  ≥ 8, we see that  +  with  ≥ 8 is the only graph in () that has the third largest closeness, which is equal to  2 +3 4 .Finally, let  ∈ () ∖ { ,3 ,  1   ,  +  } with  ≥ 8.By Lemmas 3.5 and 3.6, the maximum values of () is achieved only by one of  2  ,  * ,3 ,  ′ and  ′′ , where  ′ is obtained from  + −1 by attaching a pendant vertex at the vertex with degree two that is adjacent to a pendant vertex, and  ′′ is obtained from  * −1,3 by attaching a pendant vertex at vertex of degree three on the triangle.Note that and with  ≥ 8 is the only graph in () that has the fourth largest closeness, which is equal to  2 −+10
Let  1  ( 2  , respectively) be the graph obtained from  −1,4 by attaching a pendant vertex at a vertex of degree 2 that is adjacent (not adjacent, respectively) to the vertex of degree  − 3, see Figure 5. Lemma 3.10.Among unicyclic graphs on  vertices with girth four,  ,4 with  ≥ 5 and  1   or  * ,4 with  ≥ 6 are the only ones that have the first and the second largest closeness, which are equal to  2 +4 4 ,  2 −+9 Proof.By Lemmas 3.5 and 3.6,  ,4 is the only unicyclic graph on  vertices with girth four that has the largest closeness, and it is easy to see that ( ,4 ) =  2 +4 4 .Let  be a unicyclic graphs on  ≥ 6 vertices and the girth is four such that   ,4 .By Lemmas 3.5 and 3.6, the maximum values of () is achieved only by one of  * ,4 ,  1  ,  ′ and  2  , where  ′ is obtained from  −1,4 by attaching a pendant vertex at some pendant vertex.Then by a direct calculation, we have and for  ≥ 6.So < (4) = 0, so ( ,+1 ) < ( , ).Suppose that  is even with  ≥ 6, so ( ,+1 ) < ( , ).Thus, we conclude that among unicyclic graphs on  vertices with girth at least five,  ,5 is the unique one with the largest closeness, which is equal to  2 −+10

Closeness signless Laplacian eigenvalues
Similarly to the proof of Proposition 3.1, we have the following result, where the graph is required to be regular.
So the result follows from Proposition 2.1.
Proof.Let  be the diameter of .Then 2 ≤  ≤  − 1. Suppose that  ≥ 3. Then there is an edge  that is not a pendant edge.By Theorem 4.7,  Q 1 (  ) >  Q 1 ().So, the tree with diameter two,  1,−1 , is the unique -vertex tree that maximizes the closeness signless Laplcian spectral radius.By a direct calculation, we have Proof.Let x be the Perron vector of Q( ,ℓ ) and let  =  Q 1 ( ,ℓ ).Let  and  be two vertices in  ,ℓ so that the degree of  and  are ℓ + 1 and  − ℓ − 1, respectively.By symmetry, the entries of x at all pendant neighbors of  (, respectively) have the same value, which we denote by  (, respectively).
By deleting a pendant edge at  and adding an edge between the resulted isolated vertex and  in  ,ℓ we have a graph that is isomorphic to  ,ℓ−1 .By Rayleigh's principle, we have 1 2 ) ) By deleting  − 2ℓ − 1 pendant edges at  and adding edges between  and the resulted isolated vertices, we have a graph that is isomorphic to  ,ℓ−1 .Similarly as above, we have 1 2( − 2ℓ − 1) ) )︂ (  + ) 2 + (ℓ − 1) ) Case 1.   ≥   .

Concluding remarks
In [1], a number of results have been obtained to connect the spectral properties of the closeness matrix and the structural properties of graphs.In this paper, various connections between the spectral properties of closeness Laplacian (closeness signless Laplacian, respectively) and structural properties of graphs are established, and extremal problems to minimize certain closeness Laplacian (closeness signless Laplacian, respectively) eigenvalues are investigated.The two Laplacians based on closeness may be studied for any graphs, while the distance versions applied only to connected graphs, see [12].As compared to the ordinary Laplacian and signless Laplacian based on adjacency, the versions considered in this article also have merits as distances should be considered so as to reveal more elusive connections between spectral and structural properties.There are lots of problems to further study.For example, one may consider more extremal problems for different graph classes, and the corrections between the largest closeness Laplacian and closeness signless Laplacian eigenvalues and other distance-based graph invariants such as radius, diameter, average distance, average eccentricity, remoteness and proximity, see, e.g.[15,16].As in [17], one may also merge the spectral properties of closeness matrix and its signless Laplacian.