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MINIMUM DEGREE ENERGY of GRAPHS Dedicated to the Platinum Electronic Journal of Mathematical Analysis and Applications Vol. 7(2) July 2019, pp. 230-243. ISSN: 2090-729X(online) http://math-frac.org/Journals/EJMAA/ |||||||||||||||||||||||||||||||| MINIMUM DEGREE ENERGY OF GRAPHS B. BASAVANAGOUD AND PRAVEEN JAKKANNAVAR Dedicated to the Platinum Jubilee year of Dr. V. R. Kulli Abstract. Let G be a graph of order n. Then an n × n symmetric matrix is called the minimum degree matrix MD(G) of a graph G, if its (i; j)th entry is minfdi; dj g whenever i 6= j, and zero otherwise, where di and dj are the degrees of ith and jth vertices of G, respectively. In the present work, we obtain the characteristic polynomial of the minimum degree matrix of graphs obtained by some graph operations. In addition, bounds for the largest minimum degree eigenvalue and minimum degree energy of graphs are obtained. 1. Introduction Throughout this paper by a graph G = (V; E) we mean a finite undirected graph without loops and multiple edges of order n and size m. Let V = V (G) and E = E(G) be the vertex set and edge set of G, respectively. The degree dG(v) of a vertex v 2 V (G) is the number of edges incident to it in G. The graph G is r-regular if and only if the degree of each vertex in G is r. Let fv1; v2; :::; vng be the vertices of G and let di = dG(vi). Basic notations and terminologies can be found in [8, 12, 14]. In literature, there are several graph polynomials defined on different graph matrices such as adjacency matrix [8, 12, 14], Laplacian matrix [15], signless Laplacian matrix [9, 18], seidel matrix [5], degree sum matrix [13, 19], distance matrix [1] etc. The purpose of this paper is to study the characteristic polynomial of the minimum degree matrix and to obtain bounds for the largest minimum degree eigenvalue and minimum degree energy. Note that there are several matrices associated with graphs. In general, we can say M(G) is a matrix defined on a graph G, whose elements are given by Mij = X (di; dj) for R(i; j); where X (di; dj) is a function on degree of vertices vi and vj while R(i; j) is the relation between the vertices vi and vj. Suppose x1; x2; :::; xn are the eigenvalues 2010 Mathematics Subject Classification. 05C50. Key words and phrases. Minimum degree matrix, Minimum degree polynomial, Eigenvalues, Graph operations. Submitted April 21, 2018. Revised June 24, 2018. 230 EJMAA-2019/7(2) MINIMUM DEGREE ENERGY OF GRAPHS 231 of the matrix M, then the corresponding energy can be defined as n X EM (G) = jxij: i=1 The most extensively studied such matrix is the Adjacency matrix [12] A(G), where 1 if v v 2 E(G); A = i j ij 0 otherwise: In which, 1 for R(i; j) = vivj 2 E(G); X (di; dj) = 0 for R(i; j) = vivj 2= E(G) or vi = vj: Recently, the analogous concepts of degree sum matrix [19], degree exponent ma- trix [20] etc., were put forward. The energy EA(G) with respect to adjacency marix is smaller than the energy with respect to any other matrix. The degree exponent energy EDE(G) is larger among all other energies defined so far. The minimum de- gree energy EMD(G) is much closer to the energy EA(G) with respect to adjacency matrix, EMD(G) lies between EA(G) and EDE(G). For a complete graph Kn of n−1 order n, EMD(Kn) = (n − 1)EA(Kn) and EDE(Kn) = (n − 1) EA(Kn). These observations motivated us to study the minimum degree energy. The minimum de- gree matrix of a graph G of order n is an n×n symmetric matrix MD(G) = [mdij], whose elements are defined as minfd ; d g if i 6= j; md = i j ij 0 otherwise: Let I be the identity matrix and J be the matrix whose all entries are equal to 1. The minimum degree polynomial of a graph G is defined as PMD(G; ξ) = det(ξI − MD(G)): The eigenvalues of the matrix MD(G), denoted by ξ1; ξ2; :::; ξn are called the minimum degree eigenvalues of G and their collection is called the minimum degree spectra of G. It is easy to see that, if G is an r-regular graph, then MD(G) = rJ−rI. Therefore, for an r-regular graph G of order n, we have n−1 PMD(G; ξ) = [ξ − r(n − 1)][ξ + r] : (1.1) Let G be a graph of order n with minimum degree eigenvalues ξ1; ξ2; :::; ξn. Then the minimum degree energy EMD(G) of a graph G is defined as n X EMD(G) = jξij: i=1 Example 1. Let G = K2 · K3 be a graph (see Figure 1). Then we have the minimum degree matrix: 2 0 1 1 1 3 6 1 0 2 2 7 MD(G) = 6 7 ; 4 1 2 0 2 5 1 2 2 0 4 2 minimum degree polynomial: PMDp(G; ξ) = ξ − 15pξ − 28ξ − 12, minimum degree eigenvalues: 2 + 7; −2; −2; 2 − 7 and the minimum degree energy: EMD(G) = 8: 232 B. BASAVANAGOUD AND PRAVEEN JAKKANNAVAR EJMAA-2019/7(2) Figure 1. Example 2. Let H = C4 (see Figure 1) be a 2-regular graph. Then we have the minimum degree matrix: 2 0 2 2 2 3 6 2 0 2 2 7 MD(H) = 6 7 ; 4 2 2 0 2 5 2 2 2 0 4 2 minimum degree polynomial: PMD(H; ξ) = ξ − 24ξ − 64ξ − 48; minimum degree eigenvalues: 6; −2; −2; −2 and minimum degree energy: EMD(H) = 2r(n − 1) = 12, where r is the degree of the vertices in H. 2. Minimum degree polynomial of graphs obtained by graph operations In this section, we obtain the minimum degree polynomial of graphs obtained by some graph operations. The line graph [12] L(G) of a graph G is a graph whose vertex set is one-to- one correspondence with the edge set of the graph G and two vertices of L(G) are adjacent if and only if the corresponding edges are adjacent in G. The kth iterated line graph [6, 7, 12] of G is defined as Lk(G) = L(Lk−1(G)), k = 1; 2; :::; where L0(G) =∼ G and L1(G) =∼ L(G). k Theorem 2.1. Let G be an r-regular graph of order n and nk be the order of L (G). Then the minimum degree polynomial of Lk(G), k = 1; 2; ::: is k k k+1 nk−1 k k+1 PMD(L (G); ξ) = [ξ + 2 r − 2 + 2] [ξ − (2 r − 2 + 2)(nk − 1)]: Proof. The line graph of a regular graph is a regular graph. In particular, the line graph of a regular graph G of order n and of degree r is a regular graph of order 1 k n1 = 2 nr and degree r1 = 2r − 2 [6, 7]. Thus, the order and degree of L (G) are k−1 n Y n = (2ir − 2i+1 + 2) and r = 2kr − 2k+1 + 2: k 2k k i=0 Hence the result follows from (1.1). We use the following lemma in order to prove the following theorems. EJMAA-2019/7(2) MINIMUM DEGREE ENERGY OF GRAPHS 233 Lemma 2.2. [20] If a; b; c and d are real numbers, then the determinant of the form (ξ + a)In1 − aJn1 −cJn1×n2 (2.1) −dJn2×n1 (ξ + b)In2 − bJn2 of order n1 + n2 can be expressed in the simplified form as n1−1 n2−1 (ξ + a) (ξ + b) f[ξ − (n1 − 1)a][ξ − (n2 − 1)b] − n1n2cdg: The subdivision graph [12] S(G) of a graph G is the graph obtained by inserting a new vertex on each edge of G. Theorem 2.3. Let G be an r-regular graph of order n and size m. Then m−1 n−1 2 PMD(S(G); ξ) = (ξ + 2) (ξ + r) fξ − [(n − 1)r + 2(m − 1)]ξ +2(n − 1)(m − 1)r − [minf2; rg]2mng: Proof. Let G be an r-regular graph of order n. Then the subdivision graph of the graph G has two types of vertices. The n vertices are of degree r and the remaining m vertices are of degree 2. Hence r(J − I ) minf2; rgJ MD(S(G)) = n n n×m : minf2; rgJm×n 2(Jm − Im) Therefore, PMD(S(G); ξ) = jξI − MD(S(G))j (ξ + r)In − rJn −minf2; rgJn×m = : −minf2; rgJm×n (ξ + 2)Im − 2Jm) Using Lemma 2.2, we get the required result. The semitotal point graph [21] T2(G) is a graph which is obtained from the graph G by inserting a vertex corresponding to each edge of G and by joining each new vertex to the end vertices of the edge corresponding to it. Theorem 2.4. Let G be an r-regular graph of order n and size m. Then n−1 m−1 2 PMD(T2(G); ξ) = (ξ+2r) (ξ+2) f[ξ−2(n−1)r][ξ−2(m−1)]−[minf2; 2rg] mng: Proof. Let G be an r-regular graph of order n: Then the semitotal point graph of the graph G has two types of vertices. The n vertices are of degree 2r and the remaining m vertices are of degree 2. Hence 2r(Jn − In) minf2; 2rgJn×m MD(T2(G)) = : minf2; 2rgJm×n 2(Jm − Im) Therefore, PMD(T2(G); ξ) = jξI − MD(T2(G))j (ξ + 2r)In − 2rJn −minf2; 2rgJn×m = : −minf2; 2rgJm×n (ξ + 2)Im − 2Jm) Using Lemma 2.2, we get the required result.
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