Let $G=(V,E)$ be a simple undirected graph with n vertices and m edges. A clique in G is a subset $C\subseteq V$ such that all vertices in C are adjacent. An edge clique cover F of G is a set of cliques $F=\{C_1,C_2,\dots ,C_k\}$ that together contain each edge of G at least once. The smallest size of an edge clique cover of G is called the edge clique cover number of G and is denoted by $cc\left(G\right)$. An edge clique cover of G with size $cc\left(G\right)$ is referred to as a minimum edge clique cover of G. A special case of an edge clique cover in which every edge belongs to exactly one clique is called a clique partition of G. The size of the smallest clique partition of G is called the clique partition number of G, and is denoted by $cp\left(G\right)$. A clique partition of G with size $cp\left(G\right)$ is referred to as a minimum clique partition of G. It is clear that both $cc\left(G\right)$ and $cp\left(G\right)$ exist as E forms a clique partition (and hence an edge clique cover) of G. Further note that any minimum clique partition does not contain any cliques of size one, and, by convention, the clique partition number of the empty graph is defined to be zero. Information concerning clique partitions and edge clique covers of a graph can be found in the works [8,14,27,30].

Before defining the various matrices associated with a graph, we make note of the standard matrix notations: $I_n$ to denote the $n\times n$ identity matrix; O to denote the zero matrix (size determined by context); J to denote the all ones matrix (size determined by context); and $II$ to denote the all ones vector (size determined by context).

Given a graph G with $V=\{1,2,\dots ,n\}$ and $E=\{e_1,e_2,\dots ,e_m\}$, the (vertex-edge) incidence matrixM of G is the $n\times m$ matrix defined as follows: the rows and the columns of M are indexed by V and E, respectively; and the $(i,j)$-entry of M is 0 if $i\notin e_j$ and 1 otherwise. Similarly, the adjacency matrix $\mathcal{A}=\mathcal{A}\left(G\right)=\left(a_{ij}\right)$ is a $(0,1)$-matrix of G such that $a_{ij}=1$ if $ij\in E\left(G\right)$ and 0 otherwise. It is well-known that [18] where $D\left(G\right)$ is the diagonal matrix of vertex degrees ($d_i=deg\left(i\right)$, $i=1,2,\dots ,n$) and the matrix $Q\left(G\right)=D\left(G\right)+\mathcal{A}\left(G\right)$ is known as the signless Laplacian matrix of the graph G; the line graph, $L_G$, of the graph G is the graph whose vertex set is in one-to-one correspondence with the set of edges of G, where two vertices of $L_G$ are adjacent if and only if the corresponding edges in G have a vertex in common [22]. Finally, the equations in (1) imply an important spectral relation between the signless Laplacian matrix $Q\left(G\right)$ and $\mathcal{A}\left(L_G\right)$, see Lemma 6.

As we are also interested in studying more general symmetric matrices associated to a graph on n vertices, we let $S\left(G\right)$ denote the collection of real symmetric matrices $A=\left(a_{ij}\right)$ such that for $i\ne j$, $a_{ij}\ne 0$ if and only if $ij\in E\left(G\right)$. The main diagonal entries of any such A in $S\left(G\right)$ are not constrained. Observe that for any graph G, both $Q\left(G\right)$ and $\mathcal{A}\left(G\right)$ belong to $S\left(G\right)$.

We denote the spectrum of A, i.e., the multiset of eigenvalues of A, by $\mathrm{Spec}\left(A\right)$. In particular, where the distinct eigenvalues of A are given by ${\lambda }_1<{\lambda }_2<\cdots <{\lambda }_q$ with corresponding multiplicities of these eigenvalues are $m_1,m_2,\dots ,m_q$ respectively. Further we consider the ordered multiplicity list of A as the sequence $m\left(A\right)=(m_1,m_2,\dots ,m_q)$. For brevity, a simple eigenvalue ${\lambda }_k^{\left[1\right]}$ is simply denoted by ${\lambda }_k$.

Given a graph G, the spectral invariant $q\left(G\right)$ is defined as follows: where $q\left(A\right)$ is the number of distinct eigenvalues of A (see [2,25]). The spectral invariant $q\left(G\right)$ is called the minimum number of distinct eigenvalues of the graph G. The class of matrices $S\left(G\right)$ has been of interest to many researchers recently (see [15,16,17,17] and the references therein), and there has been considerable development on the inverse eigenvalue problem for graphs (see [23]) which continues to receive considerable and deserved attention, as it remains one of the most interesting unresolved issues in combinatorial matrix theory. Recently, J. Ahn et al. [3] offered a complete solution to the ordered multiplicity inverse eigenvalue problem for graphs on six vertices.

Using the notions of clique partitions and edge clique covers of a graph we generalize the conventional vertex-edge incidence matrix M by considering a new incidence matrix called the vertex-clique incidence matrix of a graph. Suppose $F=\{C_1,C_2,\dots ,C_k\}$ is an edge clique cover of a graph G with $V=\{1,2,\dots ,n\}$. The vertex-clique incidence matrix $M_F$ of G associated with the edge clique cover F is defined as follows: the $(i,j)$-entry of $M_F$ is real and nonzero if and only if the vertex i belongs to the clique $C_j\in F$. In the particular case when F is actually a clique partition, the vertex-clique incidence matrix, in this case, is denoted by ${\mathcal{M}}_F$, and the $(i,j)$-entry of ${\mathcal{M}}_F$ is equal to one if and only if the vertex i belongs to the clique $C_j\in F$. We observe that for any graph G the vertex-clique incidence matrix corresponding to a clique partition F, preserves several main properties of its vertex-edge incidence matrix. For instance, in Section 3, ${\mathcal{M}}_F{\mathcal{M}}_F^T={\mathcal{D}}_F+\mathcal{A}$, where ${\mathcal{D}}_F=diag(t_1^F,t_2^F,\dots ,t_n^F)$ with $t_i^F\le d_i$, where both sequences $t_i^F$ and $d_i$ are in non-increasing order. This fact enables us to determine new lower bounds for the negative eigenvalues of the graph.

The paper is organized as follows. In Section 2, we provide the necessary notions, notations, and known results that are needed in the sections containing our main observations. In Section 3, using the notion of a clique partition F of a graph G, we define signless Laplacian matrix of the graph G associated with the clique partition F. A new graph $P_G$ is introduced as a generalization for the line graph of G. In SubSection 3.1, applying this new theory of a vertex-clique incidence matrix, we produce lower bounds for the negative eigenvalues of the graph. Moreover, we present lower bounds for the negative inertia ${\nu }^{-}\left(G\right)$ of a graph G in terms of its order n and the rank of its vertex-clique incidence matrix. We also provide a sufficient condition under which the well-known inequality ${\nu }^{-}\left(G\right)\le n-\alpha \left(G\right)$ holds with equality, where $\alpha \left(G\right)$ is the independence number of G. In SubSection 3.2, we introduce new graph energies associated with a clique partition F of the graph G and study several associated properties. Moreover, new upper bounds for the energies of the graph G and its clique partition graph and line graph are determined. In Section 4, studies on the vertex-clique incidence matrix of a graph associated with an edge clique cover lead to a derivation of some new classes of graphs with $q\left(G\right)=2$ (see also SubSection 4.1).

In this section, we list some known notions, notations, and results that are needed in the remaining sections.

We start this section by introducing the notion of the eigenvalues of a graph. The eigenvalues ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ of the adjacency matrix $\mathcal{A}\left(G\right)$ (or shortened to $\mathcal{A}$ when reference to the graph G is clear from context) of the graph G are also called the eigenvalues of G. The number of positive (negative) eigenvalues in the spectrum of the graph G is called the positive (negative) inertia of the graph G, and is denoted by ${\nu }^{+}\left(G\right)$ (${\nu }^{-}\left(G\right)$). The energy of the graph G is defined as

Further details on various properties of graph energy can be found in [19,20,24,28,29]. Suppose $q_1,q_2,\dots ,q_n$ be the eigenvalues of the matrix $Q\left(G\right)$. Then the signless Laplacian energy of the graph G is defined as [1]

More information on properties of the signless Laplacian energy can be found in [1], and the energy of a line graph and its relations with other graph energies are studied in [12,21].

A subgraph H of a graph G is a graph whose vertex set and edge set are subsets of those of G. If H is a subgraph of G, then G is said to be a supergraph of H. The subgraph of G obtained by deleting either a vertex v of G or an edge e of G is denoted by $G-v$ and $G-e$, respectively. Suppose H is a graph on n vertices. Then we let $K_n\setminus H$ denote the graph obtained from the complete graph, $K_n$, by removing the edges from H. An independent set in the graph G is a set of vertices in G, no two of which are adjacent. The independence number$\alpha \left(G\right)$ of G is the number of vertices in a largest independent set of G. A matching in a graph G, is simply a collection of independent edges from G (i.e., no two edges in a matching share a common vertex from G). Additionally, a matching is referred to as perfect if each vertex from G is incident with one edge from the matching.

An $n\times n$ symmetric real matrix B is a positive semi-definite matrix if all of its eigenvalues are nonnegative. In this case, we denote $B\ge 0$. For real symmetric matrices B and C, if $B-C\ge 0$, then we write $B\ge C$.

The following result was obtained in [18].

Let ∘ denote the Schur (also known as the Hadamard or entry-wise) product. The $n\times n$ symmetric matrix A has the Strong Spectral Property (or A has the SSP for short) if the only symmetric matrix X satisfying $A\circ X=O$, $I\circ X=O$ and $[A,X]=AX-XA=O$ is $X=O$ (see [4]). The following result is given in [4].

Given two graphs G and H, the join of G and H, denoted by $G\vee H$, is the graph obtained from $G\cup H$, by adding all possible edges between G and H. Suppose G is a graph with $q\left(G\right)=2$. Then among all matrix realizations A in $S\left(G\right)$ with two distinct eigenvalues, we define the multiplicity bi-partition $[n-k,k]$ associated to A if the two eigenvalues of A has respective multiplicities $n-k$ and k. Further we define the minimal multiplicity bi-partition $MB\left(G\right)$ to be the least integer $k\le \lfloor \frac{n}{2}\rfloor$ such that G achieves the multiplicity bi-partition $[n-k,k]$. We close this section with two useful results concerning specific classes of graphs realizing two distinct eigenvalues with respect to the set $S\left(G\right)$.

In this section, we use of the vertex-clique incidence matrix associated with a clique partition of a graph G. Recall that for a graph $G=(V,E)$ with the vertex set $V=\left[n\right]=\{1,2,\dots ,n\}$ and $m=\left|E\right|$ edges, and for a given clique partition $F=\{C_1,C_2,\dots ,C_k\}$ of G, consider the matrix ${\mathcal{M}}_F$ with rows and columns indexed by the vertices in V and the cliques in F, respectively, such that the $(i,j)$-entry of ${\mathcal{M}}_F$ is equal to one if and only if the vertex i belongs to the clique $C_j\in F$. Observe that when $F=E$, ${\mathcal{M}}_F$ as simply the conventional incidence matrix of the graph G. For each vertex $i\in \left[n\right]$ of the graph G, we define a new parameter $t_i^F=t_i^F\left(G\right)$ to be the number of cliques in F containing the vertex i, that is,

We call $t_i^F\left(G\right)$the clique-degree of the vertex i in graph G associated with F, and, without loss of generality, we assume that $t_1^F\ge t_2^F\ge \dots \ge t_n^F$. Given clique partition $F=\{C_1,C_2,\dots ,C_k\}$ of G, we consider different possible classes of graphs as follows:

$\left(i\right)$ The graph G is t clique-regular if $t_1^F=\cdots =t_n^F=t$,

$\left(ii\right)$ The graph G is s clique-uniform if $|C_1|=\cdots =|C_k|=s$,

$\left(iii\right)$ The graph G is $(s,t)$ regular if $t_1^F=\cdots =t_n^F=t$ and $|C_1|=\cdots =|C_k|=s$.

Any graph is 2 clique-uniform and any d-regular graph is also d clique-regular using the trivial clique partition $F=E$.

Let ${\mathcal{D}}_F$ be the $n\times n$ diagonal matrix with row and column indexed by the vertex set V with $(i,i)$-entry equal to $t_i^F$, that is, ${\mathcal{D}}_F=diag(t_1^F,\dots ,t_n^F)$. The inner product of any two distinct rows of ${\mathcal{M}}_F$ indexed by vertices i and j is equal to the number of cliques in F containing the vertices i and j. By definition of the clique partition F, if i and j are adjacent, then this number is equal to 1 and otherwise 0. This leads to the following result:

As mentioned above, in the case of $F=E$, the matrix ${\mathcal{M}}_F$ is the incidence matrix M of G and consequently, ${\mathcal{M}}_F{\mathcal{M}}_F^T=M{M}^T$ is the signless Laplacian matrix of G, where we assume that the sequence of vertex degrees is ordered as $d_1\ge d_2\ge \cdots \ge d_n$. Notice that in this case, $t_i^F=d_i$ for $1\le i\le n$. Motivated by this observation, for any clique partition F we call ${\mathcal{Q}}_F={\mathcal{M}}_F{\mathcal{M}}_F^T$ the signless Laplacian matrix of the graph G associated with the clique partition F. Since we always have $D\ge {\mathcal{D}}_F$, it follows $Q=D+\mathcal{A}\ge {\mathcal{D}}_F+\mathcal{A}={\mathcal{Q}}_F\ge 0$. Now define the clique partition graph$P_G$ with k vertices, where each vertex i corresponds to each clique $C_i$ in F such that each pair of vertices of $P_G$ are adjacent if and only if the corresponding cliques in F have a vertex in common. If $F=E$, then $P_G=L_G$, that is, the line graph of G. The inner product of two columns of ${\mathcal{M}}_F$ is nonzero if and only if the corresponding cliques have a common vertex. From the definition of a clique partition, this nonzero value must be 1. These facts immediately yield the following result:

For the case of $F=E$, we have ${\mathcal{M}}_F^T{\mathcal{M}}_F=M^{T}M=2I_m+\mathcal{A}\left(L_G\right)$, and $P_G=L_G$ so $s_i^F=2$ for $1\le i\le k=m$.