The Hosoya index \(z(G)\) of a graph \(G\) is defined as the total number of matchings of \(G\), and the Merrifield-Simmons index \(i(G)\) of a graph \(G\) is defined as the total number of independent sets of \(G\). Although there are many known results on these two indices, there exist few on a given class of graphs with perfect matchings. In this paper, we first introduce two new strengthened transformations. Then we characterize the extremal unicyclic graphs with perfect matching which have minimal, second minimal Hosoya index, and maximal, second maximal Merrifield-Simmons index, respectively.

The aim of this paper is to introduce the notions of \(f\)-derivation and symmetric bi-derivation in \(c\)-subtraction algebras and to study some properties of these derivations.

A book-embedding of a graph \(G\) consists of placing the vertices of \(G\) on a spine and assigning edges of the graph to pages so that edges assigned to the same page without crossing. In this paper,we propose schemes to embed the connected triple-loop networks with even cardinality in books, then we give upper bounds of page number of some multi-loop networks.

Determining the size of a maximum independent set of a graph \(G\), denoted by \(\alpha(G)\), is an NP-hard problem. Therefore, many attempts are made to find upper and lower bounds, or exact values of \(\alpha(G)\) for special classes of graphs. This paper is aimed towards studying this problem for the class of generalized Petersen graphs. We find new upper and lower bounds and some exact values for \(\alpha(P(n,k))\). With a computer program, we have obtained exact values for each \(n 2k\). We prove this conjecture for some cases. In particular, we show that if \(n > 3k\), the conjecture is valid. We checked the conjecture with our table for \(n < 78\) and found no inconsistency. Finally, we show that for every fixed \(k\), \(\alpha(P(n,k))\) can be computed using an algorithm with running time \(O(n)\).

In this paper we give the solutions of finding maximum packings and minimum coverings of \(\lambda\)-fold complete symmetric digraphs with \(6\)-circuits.

In recent researches on a discriminant for polynomials, I faced a recursive (combinatorial) sequence \(\lambda_{n,m}\) whose first four terms and identities are \(\lambda_{0,m} := \binom{m}{0}\), \(\lambda_{1,m} := \binom{m}{1}=\binom{m}{m-1}\), \(\lambda_{2,m} := {\binom{m}{2}}^2 – \binom{m}{2}=\binom{m+1}{m-1}\), and \(\lambda_{3,m} = {\binom{m}{1}}^3 – 2\binom{m}{1}\binom{m}{2} + \binom{m}{3}=\binom{m+2}{m-1}\). In this paper, I introduce this sequence, prove an identity concerning it, and leave a problem and a conjecture regarding its properties.

Let \(\mu_1, \mu_2, \ldots, \mu_n\) be the eigenvalues of the sum-connectivity matrix of a graph \(G\). The sum-connectivity spectral radius of \(G\) is the largest eigenvalue of its sum-connectivity matrix, and the sum-connectivity Estrada index of \(G\) is defined as \(\mathrm{SEE}(G) = \sum_{i=1}^{n} e^{\mu_i}\). In this paper, we obtain some results about the sum-connectivity spectral radius of graphs. In addition, we give some upper and lower bounds on the sum-connectivity Estrada index of graph \(G\), as well as some relations between \(\mathrm{SEE}\) and sum-connectivity energy. Moreover, we characterize that the star has maximum sum-connectivity Estrada index among trees on \(n\) vertices.

Let \(G(n;\theta_{2k+1})\) denote the class of non-bipartite graphs on \(n\) vertices containing no \(\theta_{2k+1}\)-graph and let \(f(n; \theta_{2k+1}) = \max\{\varepsilon(G) : G \in \mathcal{G}(n;\theta_{2k+1})\}\). In this paper, we determine \(f(n; 0_5)\), by proving that for \(n \geq 11\), \(f(n; 0_5) \leq \lfloor\frac{(n-1)^2}{4}\rfloor + 1\). Further, the bound is best possible. Our result confirms the validity of the conjecture made in [1], “Some extremal problems in graph theory”, Ph.D. thesis, Curtin University of Technology, Australia (2007).

Let \(G\) be a cactus, where all blocks of \(G\) are either edges or cycles. Denote \(\mathcal{G}(n,r)\) the set of cactuses of order \(n\) and with \(r\) cycles. In this paper, we present a unified approach to the extremal cactuses for the Schultz and the modified Schultz indices.

In this paper, I study the Eulerian numbers \((A(m,k))_{k=1}^{m}\) and prove the relationship between \(\sum_{i=1}^{n}{i^m}\) and \((A(m,k))_{k=1}^{m}\), to be \(\sum_{i=1}^{n}{i^m} = \sum_{k=1}^m A(m,k)\binom{m+k}{m+1}\).