The distance spectral radius of a connected graph \(G\), denoted by \(\rho(G)\), is the maximal eigenvalue of the distance matrix of \(G\). In this paper, we find a sharp lower bound as well as a sharp upper bound of \(\rho(G)\) in terms of \(\omega(G)\), the clique number of \(G\). Furthermore, both extremal graphs are uniquely determined.

Let \(G\) be a graph with \(n\) vertices. The vertex matching polynomial \(M_v(G, x)\) of the graph \(G\) is defined as the sum of \((-1)^rq_v(G,r)x^{n-r}\), in which \(q_v(G,r)\) is the number of \(r\)-vertex independent sets. In this paper, we extend some important properties of the matching polynomial to the vertex matching polynomial \(M_v(G,2x)\). The matching and vertex matching polynomials of some important class of graphs and some applications in nanostructures are presented.

In \([18]\), Farrell and Whitehead investigate circulant graphs that are uniquely characterized by their matching and chromatic polynomials (i.e., graphs that are “matching unique” and “chromatic unique”). They develop a partial classification theorem, by finding all matching unique and chromatic unique circulants on \(n\) vertices, for each \(n \leq 8\). In this paper, we explore circulant graphs that are uniquely characterized by their independence polynomials. We obtain a full classification theorem by proving that a circulant is independence unique if and only if it is the disjoint union of isomorphic complete graphs.

We present a formula for the number of line segments connecting \(q+1\) points of an \(n_1 \times \cdots \times n_k\) rectangular grid. As corollaries, we obtain formulas for the number of lines through at least \(k\) points and, respectively, through exactly \(k\) points of the grid. The well-known case \(k = 2\) is thus generalized. We also present recursive formulas for these numbers assuming \(k = 2, n_1 = n_2\). The well-known case \(q = 2\) is thus generalized.

Let \(H\) and \(G\) be two graphs, where \(G\) is a simple subgraph of \(H\). A \(G\)-decomposition of \(H\), denoted by \((H,G)\)-GD, is a partition of all the edges of \(H\) into subgraphs (\(G\)-blocks), each of which is isomorphic to \(G\). A large set of \((H, G)\)-GD, denoted by \((H,G)\)-LGD, is a partition of all subgraphs isomorphic to \(G\) of \(H\) into \((H,G)\)-GDs. In this paper, we determine the existence spectrums for \((\lambda K_{m,n}, P_3)\)-EGD and \((\lambda K_{n,n,n}, P_3)\)-LGD.

The support of a \(t\)-design is the set of all distinct blocks in the design. The notation \(t-(v,k, \lambda|b^*)\) is used to denote a \(t\)-design with precisely \(b^*\) distinct blocks. We present some results about the structure of support in \(t\)-designs. Some of them are about the number and the range of occurrences of \(i\)-sets (\(1 \leq i \leq t\)) in the support. A new bound for the support sizes of \(t\)-designs is presented. In particular, given a \(t-(v, k, \lambda|b^*)\) design with \(b > b_0\), where \(b\) and \(b_0\) are the cardinality and the minimum cardinality of block sets in the design, respectively, then it is shown that \(b^* \geq \lceil \frac{\lceil \frac{2b}{\lambda}\rceil +7}{2}\rceil\). We also show that when \(\lambda\) varies over all positive integers, then there is no \(t-(v,k,\lambda | b^*)\)-design with the support sizes equal to \(b^*_{min}+1, b^*_{min}+2\) and \(b^*_{min}+3\), where \(b^*_{min}\) denotes the least possible cardinality of the support sizes in this design.

We consider the questions: How many longest cycles must a cubic graph have, and how many may it have? For each \(k \geq 2\) there are infinitely many \(p\) such that there is a cubic graph with \(p\) vertices and precisely one longest cycle of length \(p-k\). On the other hand, if \(G\) is a graph with \(p\) vertices, all of which have odd degree, and its longest cycle has length \(p-1\), then it has a second (but not necessarily a third) longest cycle. We present results and conjectures on the maximum number of cycles in cubic multigraphs of girth \(2, 3, 4\), respectively. For cubic cyclically \(5\)-edge-connected graphs we have no conjecture but, we believe that the generalized Petersen graphs \(P(n, k)\) are relevant. We enumerate the hamiltonian and almost hamiltonian cycles in each \(P(n,2)\). Curiously, there are many of one type if and only if there are few of the other. If \(n\) is odd, then \(P(2n, 2)\) is a covering graph of \(P(n,2)\). (For example, the dodecahedron graph is a covering graph of the Petersen graph). Another curiosity is that one of these has many (respectively few) hamiltonian cycles if and only if the other has few (respectively many) almost hamiltonian cycles.

We study the algebraic properties of soft sets in a hypermodule structure. The concepts of soft hypermodules and soft sub-hypermodules are introduced, and some basic properties are investigated. Furthermore, we define homomorphism and isomorphism of soft hypermodules, and derive three isomorphism theorems of soft hypermodules. By using normal fuzzy sub-hypermodules, three fuzzy isomorphism theorems of soft hypermodules are established.

The Merrifield-Simmons index of a graph is defined as the total number of its independent sets, including the empty set. Recently, Heuberger and Wagner [Maximizing the number of independent subsets over trees with bounded degree, J. Graph Theory, \(58 (2008) 49-68\)] investigated the problem of determining the trees with the maximum Merrifield-Simmons index among trees of restricted maximum degree. In this note, we consider the problem of determining the graphs with the maximum Merrifield-Simmons index among connected graphs of restricted minimum degree. Let \(\mathcal{G}_\delta(n)\) denote the set of connected graphs of \(n\) vertices and minimum degree \(\delta\). We first conjecture that among all graphs in \(\mathcal{G}_\delta(n)\), \(n \geq 2\delta\), the graphs with the maximum Merrifield-Simmons index are isomorphic to \(K_{\delta,n-\delta}\) or \(C_5\). Then we affirm this conjecture for the case of \(\delta = 1, 2, 3\).