ON THE PATH ENERGY OF SOME GRAPHS

Let G be a graph with vertex set V (G) = {v1, v2, ..., vn}. We define a matrix whose (i, j)th entry is the maximum number of vertex disjoint paths between the corresponding vertices if they are adjacent and is zero otherwise. We call this matrix as path matrix of G and its eigenvalues as path eigenvalues of G. In this paper, we investigate path eigenvalues and path energy of some graphs.


INTRODUCTION
The eigenvalues of a graph G are the eigenvalues of its adjacency matrix. The spectrum of a matrix is the list of its eigenvalues together with their multiplicities. The eigenvalues of graphs have several useful properties. For undefined terminology and notations, we refer to West [5] and Varga [4]. For an extensive survey on graph spectra we refer to Brouwer A. E. [3] and Beineke L. W. [6].
We define a new matrix, called the path matrix ( [1], [2]) of a graph in the following way. Let G be a graph without loops and let V (G) = {v 1 , v 2 , ..., v n } be the vertex set of G. Define the matrix P = (p i j ) of size n × n such that We call P as Path Matrix of G. The matrix P is real and symmetric. Therefore, its eigenvalues are real. We call eigenvalues of P as path eigenvalues of G.

PATH ENERGY OBTAINED FROM SOME OPERATIONS ON GRAPHS
The ordinary energy ( [7], [8]), E(G), of a graph G is defined to be the sum of the absolute values of the ordinary eigenvalues of G. Recently much work on ordinary graph energy appeared in the mathematical literature. In analogy, the path energy [2], PE(G) is defined as the sum of the absolute values of the path eigenvalues λ 1 , λ 2 , ..., λ n of G, i.e., We know if G is a r-regular, r-connected graph with n vertices, then its path matrix has row sum (n − 1)r and this row sum (n − 1)r is one of the path eigenvalues of G and the other path eigenvalues are −1 with multiplicity n − 1.
In the following Theorem, we investigate the path eigenvalues and path energy of a graph which is obtained by joining a vertex of r 1 -regular, r 1 -connected graph with a vertex of r 2regular, r 2 -connected graph by an edge. Proof. Let P, Q, and R be the path matrices of G, G 1 , and G 2 respectively. As G 1 is r 1 -regular, r 1 -connected with m vertices, the path eigenvalues of G 1 are λ 1 = (m − 1)r 1 with multiplicity 1, −r 1 with multiplicity m − 1 and as G 2 is r 2 -regular, r 2 -connected on n vertices, the path eigenvalues of G 2 are µ 1 = (n − 1)r 2 with multiplicity 1, −r 2 with multiplicity n − 1. The path matrix P can be written as where J m×n is m by n matrix with all entries 1. We know that 1 is an eigenvector of Q corresponding to (m − 1)r 1 , so we assume Thus A be a square matrix, then the sum of all 2 × 2 principal minors of A is equal to s 2 (A), where s 2 (A) is the second elementary symmetric function of the eigenvalues of A. Thus s 2 (P) = Now for the path matrices Q and R, we get Again every principal minor of size 2 × 2 of P is either a 2 × 2 principal minor of Q or R, or it has the form q ii 1 1 r j j = q ii r j j − 1, i = 1, 2, ..., m, j = 1, 2, ..., n, where Q = (q i j ) and R = (r i j ).
Using this, we write Hence We know for a tree T with m vertices, its path matrix has row sum m − 1 and this row sum m − 1 is one of the path eigenvalue of T and the other path eigenvalues are −1 with multiplicity m − 1.
In the following Proposition, we investigate the path eigenvalues and path energy of a graph which is obtained by joining a vertex of a tree with a vertex of r-regular, r-connected graph by an edge.
Proposition 2.2. Let G 1 be a tree with m vertices and G 2 be r-regular, r-connected graph with n vertices. Let G be a graph obtained by joining a vertex of G 1 to a vertex of G 2 by an edge.
Then the path eigenvalues of G are −1 with multiplicity m − 1, −r with multiplicity n − 1, Proof. Let P, Q (= J m − I m ), and R be the path matrices of G, G 1 , and G 2 respectively. Here µ 1 = (n − 1)r. The path matrix P can be written as where J m×n is m by n matrix with all entries 1. We know that 1 is an eigenvector of Q corresponding to λ 1 = (m − 1), so we assume 1X = 0, where X = [x 1 , ..., x m ] is an eigenvector of Q can write this as Now for the path matrices Q and R, we get Again every principal minor of size 2 × 2 of P is either a 2 × 2 principal minor of Q or R, or it has the form q ii 1 1 r j j = q ii r j j − 1, i = 1, 2, ..., m, j = 1, 2, ..., n, where Q = (q i j ) and R = (r i j ).
Using this, we write s 2 (P) = s 2 (Q) + s 2 (R) + ∑ m i=1 ∑ n j=1 (q ii r j j − 1) = s 2 (Q) + s 2 (R) − mn (ii) From (i) and (ii), we get Hence We investigate the path eigenvalues and path energy of a graph which is obtained by taking k copies of r-regular, r-connected graph and joining a vertex of one graph with a vertex of other graph.
Theorem 2.3. Let G 1 , G 2 , ..., G k be the k copies of some r-regular r-connected graph on n vertices and let G be a graph obtained by joining a vertex of G i with a vertex of G i+1 (1 ≤ i ≤ k − 1) by an edge. Then the path eigenvalues of G are n(k − 1) + r(n − 1) with multiplicity 1, −r with multiplicity k(n − 1) and n(r − 1) − r with multiplicity k − 1.
Proof. Let P be the path matrix of G and Q be the path matrix of G i , for i = 1, 2, ..., k. Let J n be the n × n matrix with all entries 1. The path matrix P can be written as Now subtracting the first row from 2 nd , 3 rd , ..., k th rows, we get This is a triangular block matrix. Hence the characteristic polynomial of P is C P (x) = |Q + (k −

CONCLUSION
In the present paper, path eigenvalues and path energy of graphs which are obtained by joining a vertices of some specific classes of graphs are obtained and studied.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.