Degree sum adjacency polynomial of standard graphs and graph operations

In this paper, we explore the characteristic polynomials of degree sum adjacency matrix DS A ( G ) of a simple undirected graph G . We state a relation between the structure of a graph and the coefﬁcients of its DS A polynomial. A walk generating function is expressed in terms of DS A polynomial. Then, we obtain the degree sum adjacency polynomial for some standard graphs, derived graphs and for graph operations.


Introduction
Spectral graph theory focuses on the study of the eigenvalues and its relation to the structural properties of a graph. Thus, for a given graph many matrices were defined in this field which records the information about the vertices and the edges of a graph. To state a few, the most explored and widely studied matrices are the adjacency matrix, the laplacian matrix, the signless laplacian matrix, and many more.
In chemistry, many matrices are defined with respect to the distance, incidence and other factors. This motivated many researchers to explore different matrices [11,13,14] and study their properties and energy [1,10]. Zagreb index defined as the sum of the degrees of adjacent vertices have been studied intensively [4,5,6,7,15], which relates to the degree sum adjacency (DS A ) matrix. This motivated us to explore the DS A polynomial for a graph and its operations. In this paper, we consider the degree sum adjacency matrix defined by Zaferani [14] and we discuss relation between the structure of a graph and the coefficients of DS A polynomial. Then we determine the generating function for the number of walks of each length with respect to the degree sum adjacency matrix. Later we study the DS A polynomial of complementary graphs, some regular graphs, derived graphs and graph operations in terms of its adjacency polynomial. The proof techniques of the results in this paper are analogous to the results in [3].
Let G be a simple graph with n vertices and m edges. The adjacency matrix of a graph G is defined as A(G) = [a ij ], where a ij = 1, if v i is adjacent to v j and a ij = 0 otherwise. The adjacency eigenvalues are denoted as λ 1 ≥ λ 2 ≥ · · · ≥ λ n and they satisfy all the basic relations [3]. The adjacency polynomial of a graph G is denoted by, ϕ (G : λ) = det(λI − A) = a 0 λ n + a 1 λ n−1 + · · · + a n .
Let the vertices v 1 , v 2 , . . . , v n of G have the degrees d 1 , d 2 , . . . , d n . Then DS A (G) = [ds ij ] is the degree sum adjacency matrix [14] of G whose elements are defined as, G:

Graph and its DS A matrix
The degree sum adjacency polynomial of a graph G is defined as As DS A (G) is a real symmetric matrix, its eigenvalues must be real and can be arranged as β 1 ≥ β 2 ≥ · · · ≥ β n .
The eigenvalues of matrix xI + yJ of order n × n are x + ny with multiplicity one and x with multiplicity n − 1.

Characteristic polynomial of degree sum adjacency matrix
In this section, first we obtain the explicit values of some coefficients of polynomial as defined in Eq. (2). Then obtain the relation between the DS A characteristic polynomial of a graph and that of its complement.
Some propositions relating the coefficients a i of P DS A (G) (β) to structural properties of G: A degree sum adjacency matrix of any simple graph G is, Then the coefficients of DS A polynomial of G can be expressed using Sach's theorem as follows.
Let G be a graph having n vertices and i be any positive number. Then Sach's graphs S i are the subgraphs of G with i vertices having disjoint union of K 2 and/or C n . Let the number of components of s ∈ S and number of cycles of s ∈ S be P (s) and c(s) respectively. Then the coefficient a i of β n−i in Eq. (2) is given by ) .
Here we state first few coefficients of DS A polynomial. ) .
Relation between DS A polynomial of a graph and its complement: A walk of length k in a graph is any sequence of vertices v 1 , v 2 , . . . , v k+1 (not necessarily different) such that there is an edge from v i to v i+1 , for each i = 1, 2, . . . , k. To obtain the DS A polynomial of a complement graph we first find the generating function to get the number of walks of length k in G with respect to its DS A matrix. www.ejgta.org Proof. The proof of this theorem is analogous to the proof obtained for adjacency matrix of a graph G [3]. Let sum (A) denote the sum of all entries of matrix A.
where B is any n ordered non singular matrix, J is a square matrix whose all entries are equal to one, x is any arbitrary number and N k is the number of all walks of length k in G with respect to the DS A matrix. Let H DS A (G) (t) = ∑ ∞ k=0 N k t k denote the generating function that gives the number of walks N k each of length k in G. Using Eq. (8), Eq. (7) and Eq. (6) we get.
From Eq. (5) we have Substituting x = 2rt in the Eq.(10) we get www.ejgta.org Degree sum adjacency polynomial of standard graphs and graph operations | S. S. Shinde et al. But Multiplying both sides by t we get, Using the above result in Eq.(11) we get Hence we get the required generating function.

Theorem 2.2. If G is a regular graph with degree r and n vertices, then DS
Proof. Since G is a r regular graph, a walk can begin at any one vertex of G and may continue in r ways. Therefore, number of walks of length k in G is N k = nr k .
Thus, for DS A (G) we have N k (2r) k = nr k . Hence for the generating function H G (t) we have, Using Eq.(4) we get Substituting − Simplifying we get the required DS A polynomial for G in terms of DS A polynomial of G.
3. DS A polynomials and spectra of some regular graphs Theorem 3.1. [14] The degree sum adjacency polynomial of a complete graph K n with n vertices is This result can also be obtained by using lemma (1.1).
Proof. As each component of a 1-regular graph is isomorphic to K 2 , by substituting n = 2 in Eq.
A cocktail-party graph is a complementary graph of 1-regular graph.
Corollary 3.1. The DS A -polynomial of the cocktail-party graph with 2k vertices is Proof. Let G be a 1-regular graph, then P DS A (G) = P DS A (CP (k)) . To obtain DS A polynomial for cocktail-party graph, substitute n = 2k and r = 1 in Eq. (12) Theorem 3.3. If C n is a cycle with n vertices, then eigenvalues of degree sum matrix of C n are A crown graph S 0 n is obtained from the complete bipartite graph K n,n by deleting the perfect matching edges.
Proof. The DS A -matrix of crown graph will be of the form , where X is a matrix of all zeros and Y is a matrix with all non diagonal entries as 2(n − 1) and the diagonl entries as zero. The matrix Y is of the form 2(n − 1)J − 2(n − 1)I. Separately evaluating (X − Y ) and (X + Y ) by applying lemma (1.1) and then multiplying, we get the required result.

DS A polynomial of some graph operations
Line Graph L(G) of a graph G is the graph which has one-to-one correspondence between the vertex set and the set of edges of the graph G, with two vertices of L(G) being adjacent iff the corresponding edges are adjacent in G [8].
Proof. Let A be an adjacency matrix of graph G, B be an adjacency matrix of graph L(G) and R be the incidence matrix of G with D as the degree matrix. Then for G, we have substituting r ′ (β − 2) = β and r ′ = 4r − 4 we get the required result as shown in Eq. (20).
Subdivision graph s(G) of a simple graph G is the graph which is obtained by adding (inserting) a new vertex onto every edge of G [8].
Proof. For a r−regular graph G having n vertices, its degree sum adjacency matrix of subdivision graph s(G) of graph G is DS A (s(G)). As vertex set of s(G) is partitioned into two sets, one with www.ejgta.org .
Semi total point graph T 1 (G) is a graph which is derived from graph G by inserting (adding) a new vertex into every edge of G and each new inserted vertex is then joined to the end points of the corresponding edge [3].
Theorem 4.3. The DS A polynomial P DS A (T 1 (G)) of semi total point graph T 1 (G) of a n ordered r-regular graph G in terms of its adjacency polynomial ϕ(G) is Proof. Let G be a r-regular graph with n vertices, where m = nr/2 new vertices are added to construct a T 1 (G) graph. Then the DS A polynomial of T 1 (G) is DS A (T 1 (G)) = det(βI − DS A (T 1 (G))).
Semi total line graph T 2 (G) of a graph G, is the graph with vertex set V (T 2 (G)) = V (G) ∪ E(G) in which two vertices are adjacent if they are on adjacent edges of G or one is a vertex of G and the other is an edge of G, incident to it [3].

Theorem 4.4. Let G be a r-regular graph having n vertices and m edges and let T 2 (G) be a semi total line graph of G. Then the DS A polynomial P DS A (T 2 (G)) of semi total line graph T 2 (G) of a graph G in terms of its adjacency polynomial of line graph ϕ(L(G)) is
Proof. For a r-regular graph G, the DS A polynomial of T 2 (G) is . T horn graph G +k is a graph which is obtained from graph G by attaching k pendent vertices to every edge of G. If G is a graph with n vertices and m edges, then G +k has n + nk vertices and m + nk edges.
Theorem 4.5. The DS A polynomial P DS A (G +k ) of Thorn graph G +k of a n ordered r-regular graph G in terms of its adjacency polynomial ϕ(G) is Proof. The DS A polynomial of Thorn graph can be written as, where A is the adjacency matrix of G, I is the unit matrix and J is a block matrix of order (n, k).
For β ̸ = 0, multiply the rows (consisting of block matrices) numbered 2, 3, . . . , k+1 by www.ejgta.org and add the resulting rows to the first row. This reduces the determinant as follows.
Proof. Let G be a r regular graph with n vertices and m edges. As DS A (T (G)) can be expressed in terms of its adjacency matrix A, adjacency matrix of line graph B and the incidence matrix R of a graph G, we get ) .
Its DS A polynomial can be expressed as .
Applying series of elementary transformation, • Second row = second row -R T first row • First row = First row + 4rR β + 8r second row the determinant can be expressed as follows.
where λ i (i = 1, 2, . . . , n) are the eigenvalues of A. Thus we have proved that there are exactly (m − n) DS A eigenvalues of T (G) equal to β = −8r.
Using b 2 − 4ac, we find that the roots of the polynomial aβ 2 + bβ + c where a = 1, b = −(4r 2 − 8r + 8rλ i ) and c = 16r 2 λ 2 i + λ i (16r 3 − 48r 2 ) − 16r 3 . On solving we get 2n eigenvalues of T (G) as The join G 1 ∇G 2 of (disjoint) graphs G 1 and G 2 is the graph that is obtained from G 1 ∪ G 2 , by joining every vertex of G 1 to all vertices of G 2 .
Theorem 4.7. Let G 1 and G 2 be two regular graphs with regularity r 1 and r 2 and with orders n 1 and n 2 respectively. Then the DS A -polynomial of G 1 ∇G 2 is given by the relation, where x = n 1 + n 2 + r 1 + r 2 .
Proof. The DS A -polynomial of G 1 ∇G 2 is obtained as where x = n 1 + n 2 + r 1 + r 2 and J is a matrix whose all entries are equal to unity. The above determinant can be written as, Where ds ij is the ij th entry DS A matrix of G 1 and ds ′ ij is the ij th entry DS A matrix of G 2 . In G 1 each vertex is adjacent to all vertices of G 2 , so its new vertex degree is r 1 + n 2 and as there are r 1 vertices adjacent to a vertex v i in G 1 , therefore for i = 1, 2, . . . , n 1 .
Adding first row to all other rows of the determinant, we get Similarly, we can show that from Eq. (30) we get Let G be a graph with n 1 vertices and let H be a graph with n 2 vertices. Then the corona G•H is the graph with n 1 + n 1 n 2 vertices, which is obtained by taking graph G and n copies of graph H and by joining i th vertex of G to each vertex in the i-copy of H (i = 1, · · · , n 1 ). [ β 2(r 1 + n 2 ) − m(r 1 + r 2 + n 2 + 1) 2 2(r 1 + n 2 )(β − 2(r 2 + 1)) ]) .
Theorem 4.9. The DS A -polynomial of cartesian product of complete graphs K 2 and K n , K 2 K n is Proof. As complement of 2n-vertex crown graph S 0 n is the cartesian product of K 2 and K n , K 2 K n . Applying the result of Theorem (3.4) in Eq. (12) of Theorem(2.2) we get the required result.