This paper de nes the concept of partitioned hypergraphs, and enumerates the number of these hypergraphs and discrete complete hypergraphs. A positive equivalence relation is de ned on hypergraphs, this relation establishes a connection between hypergraphs and graphs. Moreover, we de ne the concept of (extended) derivable graph. Then a connection between hypergraphs and (extended) derivable graphs was investigated. Via the positive equivalence relation on hypergraphs, we show that some special TREES are derivable graph and complete graphs are self derivable graphs.

One of the most important forest TREES in the south of Kerman, which plays an important role in the region's ecosystem and beekeeping industry and is infected by wood-eating beetles, is Prosopis cineraria. These TREES are an important habitat for various animals and refreshing the hot air of the south. In the sampling conducted during 1401-1400 from the mesquite forests of southern Kerman (Ghaleganj, Faryab, Anbarabad), the wood-eating beetle Xylopertha reflexicauda (Bostrichidae) was collected for the first time from Prosopis cineraria TREES and is identified. The species of wood-eating beetle was identified by reliable scientific sources and was finally confirmed by Mr. Dr. Len Yu Liu. This species was first identified and described by Lesne in 1937. Due to the lack of water and recent droughts and the weakness of Iranian mesquite TREES, the larvae of this beetle are active inside the trunk and bark of mesquite TREES and feed on the wood and bark of tree trunks and cause great damage to Prosopis cineraria TREES.

Based on the research, belief in sacred TREES can be found in various forms around the world and for a variety of reasons. In Iranian popular culture, some plants and TREES have been given special power, properties and sanctity. There are few places in Iran that do not have a sample of respected TREES. These different TREES are generally believed to have relatively similar characteristics, including, prohibition of cutting, prohibition of breaking their branches that may cause destruction or serious damage to one who does so. According to popular belief, these TREES meet the needs of the needy and are effective in repelling some diseases and evil eye. The results show that, in general, belief in sacred TREES is different in theistic and atheistic religions. Among Muslims, the sanctity of certain TREES is mostly related to the manifestation of God and the position of the pure souls of saints and mystics, or to their tombs, or to the deeds and life events of prophets and religious leaders, but in atheistic religions, the sanctity of TREES is reduced to the status of spirits and goddesses and in practice to the level of magic and spell. Using written sources and descriptive-analytical method, this research has been done.

Plane TREES and noncrossing TREES have been generalized by assigning labels to the vertices from a given set such that a prior coherence condition is satisfied. These TREES are called k-plane TREES and k-noncrossing TREES respectively if k labels are used. Results of plane TREES and noncrossing TREES were recently unified by considering d-dimensional plane TREES where plane TREES are 1-dimensional plane TREES and noncrossing TREES are 2-dimensional plane TREES. In this paper, d-dimensional k-plane TREES are introduced and enumerated according to number of vertices and label of the root, root degree, number of components constituting a forest, label of the eldest child of the root and the length of the leftmost path. The equivalent results for plane TREES and noncrossing TREES follow easily from our results as corollaries.

We give an alternative treatment and extension of some results of Itoh and Mahmoud on one-sided interval TREES. The proofs are based on renewal theory, including a case with mixed multiplicative and additive renewals.

‎Zagreb indices were reformulated in terms of the edge degrees instead of the vertex degrees‎. For a graph $G$‎, ‎the first and second reformulated Zagreb indices are defined respectively as‎:‎$$EM_1(G)=\sum_{\varepsilon\in E(G)}d^2(\varepsilon), EM_2(G)=\sum_{\varepsilon,\varepsilon'\in E(G),\,\varepsilon\sim \varepsilon'}d(\varepsilon)\,d(\varepsilon'),$$‎ where $d(\varepsilon)$ and $d(\varepsilon')$ denote the degree of the edges $\varepsilon$ and $\varepsilon'$ respectively‎, ‎and $\varepsilon\sim \varepsilon'$ means that the edges $\varepsilon$ and $\varepsilon'$ are adjacent‎. In this paper‎, ‎we obtain sharp lower bounds on the first and second reformulated Zagreb‎ indices with a given number of vertices and maximum degree‎. ‎Furthermore‎, ‎we will determine the extremal TREES that achieve these lower bounds‎.

Let G=(V, E) be a graph and let f: V(G)→ {0, 1, 2} be a function. A vertex v is protected with respect to f, if f(v)>0 or f(v)=0 and v is adjacent to a vertex of positive weight. The function f is a co-Roman dominating function, abbreviated CRDF if: (i) every vertex in V is protected, and (ii) each u∈ V with positive weight has a neighbor v∈ V with f(v)=0 such that the function f_uv: V→ {0, 1, 2}, defined by f_uv (v)=1, f_uv (u)=f(u)-1 and f_uv (x)=f(x)for x∈ V-\{v, u}, has no unprotected vertex. The weight of f is ω (f)=∑ _(v∈ V)▒ 〖 f(v)〗 . The co-Roman domination number of a graph G, denoted by γ _cr G), is the minimum weight of a co-Roman dominating function on G. In this paper, we first present an upper bound on the co-Roman domination number of TREES in terms of order, the number of leaves and supports. Then we find bounds on the co-Roman domination number of a graph and its other dominating parameters.