The Merrifield-Simmons index i G of a graph G is defined as the number of subsets of the vertex set, in which any two vertices are nonadjacent, that is, the number of independent vertex sets of G The Hosoya index z G of a graph G is defined as the total number of independent edge subsets, that is, the total number of its matchings. By C n, k, λ we denote the set of graphs with n vertices, k cyc...

The bull is the graph consisting of a triangle and two disjoint pendant edges. A graph is called bull-free if no induced subgraph of it is a bull. This is the first paper in a series of three. The goal of the series is to explicitly describe the structure of all bull-free graphs. In this paper we study the structure of bull-free graphs that contain as induced subgraphs three-edge-paths P and Q,...

For a graph G = (V, E), a bijection f from V(G) ∪ E(G) → {1, 2, . . . , |V(G)| + |E(G)|} is called (a, d)-edge-antimagic total ((a, d)EAT) labeling of G if the edge-weights w(xy) = f(x) + f(y) + f(xy), xy ∈ E(G), form an arithmetic progression starting from a and having a common difference d, where a > 0 and d ≥ 0 are two fixed integers. An (a, d)-EAT labeling is called super (a, d)-EAT labelin...

The reciprocal complementary Wiener number of a connected graph G is defined as ( ) { } ( ) ( ) | ∑ u v V G RCW G d d u v G ⊆ = + − , 1 1 , where ( ) V G is the vertex set. ( ) | d u v G , is the distance between vertices u and v, and d is the diameter of G. A tree is known as a caterpillar if the removal of all pendant vertices makes it as a path. Otherwise, it is called a non-caterpillar. Amo...

A k−geodominating set is a geodominating set S such that any vertex v ∈ V (G)\S is geodominated by a pair x, y of vertices of S with d(x, y) = k. A k-perfect geodominating set is a geodominating set S such that any vertex v ∈ V (G) \ S is geodominated by exactly one pair x, y of the vertices of S with d(x, y) = k. The cardinality of a minimum perfect geodominating set in G is its perfect geodom...

For a simple graph G = (V,E) with the vertex set V and the edge set E , a vertex irregular total k -labeling f : V ∪E → {1, 2, . . . , k} is a labeling ∗ Also at Department of Mathematics, Faculty of Mathematics and Sciences, University of Gadjah Mada, Yogyakarta, Indonesia. D. INDRIATI ET AL. /AUSTRALAS. J. COMBIN. 65 (1) (2016), 14–26 15 of vertices and edges of G in such a way that for any t...

The matching energy of a graph was introduced by Gutman and Wagner in 2012 and defined as the sum of the absolute values of zeros of its matching polynomial. Let θ(r, s, t) be the graph obtained by fusing two triples of pendant vertices of three paths Pr+2, Ps+2 and Pt+2 to two vertices. The graph obtained by identifying the center of the star Sn−g with the degree 3 vertex u of θ(1, g−3, 1) is ...

Submodularity is an important  property of set functions with deep theoretical results  and various  applications. Submodular systems appear in many applicable area, for example machine learning, economics, computer vision, social science, game theory and combinatorial optimization.  Nowadays submodular functions optimization has been attracted by many researchers.  Pendant pairs of a symmetric...

In this paper, we investigate the existence of nontrivial solutions for the equation γ(G□H) = γ(G) γ(H) fixing one factor. For the complete bipartite graphs Km,n; we characterize all nontrivial solutions when m = 2, n ≥ 3 and prove the nonexistence of solutions when m, n ≥ 3. In addition, it is proved that the above equation has no nontrivial solution if H is one of the graphs obtained from Cn,...

A caterpillar graph is a tree in which the removal of all pendant vertices results in a chordless path. In this work, we determine the number of maximal independent sets (mis) in caterpillar graphs. For a general graph, this problem is #P—complete. We provide a polynomial time algorithm to generate the whole family of mis in a caterpillar graph. We also characterize the independent graph (inter...