A New Method for the Sum-Edge Characteristic Polynomials of Graphs

In this paper, the determinant of the sum-edge adjacency matrix of any given graph without loops is calculated by means of an algebraic method using spanning elementary subgraphs and also the coefficients of the corresponding sum-edge characteristic polynomial are determined by means of the elementary subgraphs. Also, we provide a formula for calculating the number of smallest odd-sized cycles in a given regular graph.


Introduction
Let G be an arbitrary graph with no loops. A lot of algebraic, physical and chemical properties of a given graph G can be determined by means of matrices corresponding to G. The most famous such matrices are the adjacency, incidency and Laplacian matrices. For example, the intermolecular energy of a chemical compound can be found only by mathematical calculations on the graph modelling the chemical compound as the sum of the absolute values of the eigenvalues of the adjacency matrix, see [4], [5], [7] and [12]. There are some other graph matrices with several uses, see e.g. [2] and [11]. Accordingly, several types of characteristic polynomials and energy are defined by means of these matrix types, [16] and [17]. One of the most popular of them is the Laplacian energy, see e.g. [8]. In [9], some degree based energies have been studied. Das studied maximum energy of bipartite graphs in [10]. In [13], energy of regular graphs is considered. Naturally, being a linear algebraic concept, graph energy is related to many other notions related to graphs. For example the relation between energy and matching number have been considered in [1] and [15]. In this paper, we study the sum-edge adjacency matrix and we determine the sum-edge characteristic polynomial.

Sum-edge adjaceny matrices, determinants and characteristic polynomials
Firstly, we recall the definition of sum-edge adjacency matrix of G, see e.g. [11]. Let the vertex set of G be V (G) = {v 1 , · · · , v n } and edge set E(G). We shall denote the sum-edge adjacency matrix of G by S(G). S(G) = [s ij ] n×n is determined by the adjacency of vertices as follows: We will call the characteristic polynomial of S(G) as the sum-edge characteristic polynomial of G and denote it by P se G (x). These polynomials for several graphs were studied in [14].
Secondly, let us take a graph G. A subgraph G ′ t of G is called an elementary subgraph if every component of G ′ t is either an edge or a cycle, see e.g. [2]. Also, if the number of vertices of an elementary subgraph is equal to the number of vertices of G, then the elementary subgraph is called a spanning elementary subgraph. c − (G ′ t ) and c • (G ′ t ) are defined as the number of components in a subgraph G ′ t which are edges and cycles, respectively.
where the summation is over all permutations i 1 i 2 · · · i n of the set S = {1, 2, · · · , n}. Since for a set with n elements, the number of permutations is given by n!, we sum all n! terms. In other words, we can express the formula as follows: where the summation is over all permutations of {1, · · · , n}. The sign is taken as + or − with respect to whether the permutation is even or odd.
Let us define new notions in preparation to the forthcoming theorems. Let u i and u j be two adjacent vertices in an edge component of corresponding elementary (spanning elementary) subgraph G ′ t of G. Similarly, let v i and v j be two adjacent vertices in a cycle component of corresponding elementary (spanning elementary) subgraph G ′ t of G. In the next theorem, we provide a formula for calculating the determinant of S(G) whose proof is completed by using a method given in [3].
where the summation is over all spanning elementary subgraphs G ′ t . Note that in a cycle component of the spanning elementary subgraph G ′ t of G, if v i and v j are two adjacent vertices in this cycle component, then when calculating the latter product in Theorem 3.2 corresponding to cycle component, we consider only one of the edges v i v j and v j v i due to the permutation structure. Let now u i and u j be two adjacent vertices in an edge component of the spanning elementary subgraph G ′ t of G. We consider both of the edges u i u j and u j u i due to the transposition structure and therefore the square appears in the first product corresponding to edge components.
If either one of the cycle components or edge components does not exist in the spanning elementary subgraph, we exclude the corresponding product in the statement of Theorem 3.2.
Proof. By the Definition 3.1, If some s iσ(i) = 0 for i = 1, · · · , n, then this means that v i and v σ(i) are not adjacent vertices. Hence, if the term corresponding to a permutation σ is non-zero, then we know that any permutation can be expressed uniquely as the product of disjoint cycles whose length at least two. Every cycle (ij) of length two corresponds to the factors s ij s ji and represents a single edge If any spanning elementary subgraph has at least two edge components, then each edge component contributes to the product by s ij s ji . Moreover, every cycle (123 · · · n) of length greater than or equal to three is in the form (v 1 v 2 · · · v n ) and corresponds to the product by s 12 s 23 · · · s n1 . This means that the contribution of the cycle components to the determinant will be [d If any spanning elementary subgraph has at least two cycle components, then each such cycle component contributes to the corresponding product by s 12 s 23 · · · s n1 . Also, any spanning elementary subgraph consists of both cycle and edge components. Then both of these component types contribute factors as we have mentioned above. Thus, as a result, every non-zero term in the determinant expansion is an element of an elementary subgraph G ′ t of G.
As we have already noted before, any permutation σ can be expressed in a unique way as the product of some disjoint cycles and in such a representation, the sign of σ will be (−1) n−the number of cycles in the cycle decomposition of σ Now, we consider cycle components in G ′ t . For each of these, we have two choices, so such permutations corresponding to cycle components contribute 2 c•(G ′ t ) . Thus, we finish the proof.  We can now give the formula for the factors of the sum-edge characteristic polynomial of any graph G by means of [2]:

Theorem 3.4. Let G be any graph of order n. Then
Here the summation is over all elementary subgraphs G ′ t with l vertices.
Proof. Every k l is calculated by means of the corresponding elementary subgraphs of G with l vertices. As a method for calculating value of k l , we have to calculate all principal minors of S(G) with l vertices. After that, we sum all of these and lastly multiply with (−1) l , for l = 1, · · · , n. By Theorem 3.2, we can write determinant formula instead of principal minors and hence, we achive the formula which is equal to the formula where the summation is over all elementary subgraphs G ′ t having l vertices. .
Then adding all these up, we get k 2 = −52.

Consequently, by Theorem 3.4,
We can now prove the following result by means of [3]: Theorem 3.6. Let G be a graph of order n. Let P se G (x) = |xI − S(G)| = x n + k 1 x n−1 + k 2 x n−2 + · · · + k n . If k 1 , k 3 , · · · , k 2m+1 = 0, then G has no odd cycle of length 2i + 1 where i = 0, 1, · · · , m. Moreover, if G is a regular graph, then the number of (2m Note that v i and v j are adjacent vertices in one of the elementary subgraphs with 2m + 3 vertices. Proof. For any graph G, the first coefficient of the characteristic polynomial of G is k 1 = 0. Since we assume that k 3 = 0, there are no 3-cycles in G. Hence, we next look for 5-cycles. If G has at least one 5-cycle, we have the result. If k 5 = 0, then there are no 5-cycles in G. We continue the process until we find all k 1 , k 3 , · · · , k 2m+1 = 0. Finally we have to obtain an elementary graph having at least one k 2m+3cycle. The elementary subgraph will have 2m + 3-cycles. Let one of these cycles be (1 2 · · · (2m + 3)). Then we have the edge decomposition v 12 v 23 · · · v (2m+3)1 . By Theorem 3.4, we have Let us apply Theorem 3.6 to the graph G given in Fig. 3.

Figure 3: Graph G
The sum-edge characteristic polynomial of the graph G is If we equalize this equation to the then by Theorem 3.6, we see that k 1 = 0, and we will look for k 3 .
where v i and v j are vertices which are adjacent to each other in the component of one of the elementary subgraphs with 3 vertices. Moreover, you can observe that the value of the subgraphs G ′ 1 , G ′ 2 and G ′ 3 are not the same.
The previous theorem is not verified because the graph G is not regular. Example 3.8. Let us apply the Theorem 3.6 to the complete graph K 4 given in Fig. 4: The sum-edge characteristic polynomial of the graph K 4 is If we equalize this equation to the then by Theorem 3.6, we see that k 1 = 0 and now we can look for k 3 .