The concept of the sum graph and integral sum graph were introduced by F. Harary. In this paper, we gain some upper and lower bounds on the sum number and the integral sum number of a graph and these bounds are sharp, and some new properties on the integral sum graph. Using these results, we could directly investigate and determine the exclusive integral sum numbers, the exclusive sum numbers, the sum numbers and the integral sum numbers of the graphs \(K_n\backslash E(2P_3)\), \(K_n\backslash E(P_3)\) and any graph \(H\) with minimum degree \(\delta(H) = n-2\) respectively as \(2\) is more than a given number. Then they will be the beginning of a new thought of research on the (exclusive) sum graph and the (exclusive) integral sum graph.

Let \(G = (V, E)\) be a simple connected graph, where \(d_u\) is the degree of vertex \(u\), and \(d_G(u, v)\) is the distance between \(u\) and \(v\). The Schultz index of \(G\) is defined as \(\mathcal{W}_+(G) = \sum\limits_{u,v \subset V(G)} (d_u + d_v)d_G(u,v).\)In this paper, we investigate the Schultz index of a class of trees with diameter not more than \(4\).

Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\), and let \(g\) and \(f\) be two integer-valued functions defined on \(V(G)\) such that \(0 \leq g(x) \leq f(x)\) for each \(x \in V(G)\). A \((g, f)\)-factor of \(G\) is a spanning subgraph \(F\) of \(G\) such that \(g(x) \leq d_F(x) \leq f(x)\) for each \(x \in V(F)\). A \((g, f)\)-factorization of \(G\) is a partition of \(E(G)\) into edge-disjoint \((g, f)\)-factors. Let \({F} = \{F_1, F_2, \ldots, F_m\}\) be a factorization of \(G\) and \(H\) be a subgraph of \(G\) with \(m\) edges. If \(F_i\), \(1 \leq i \leq m\), has exactly one edge in common with \(H\), we say that \({F}\) is orthogonal to \(H\). In this paper, it is proved that every \((mg+k-1, mf-k+1)\)-graph contains a subgraph \( {R}\) such that \( {R}\) has a \((g, f)\)-factorization orthogonal to any given subgraph with \(k\) edges of \(G\) if \(f(x) > g(x) \geq 0\) for each \(x \in V(G)\) and \(1 \leq k \leq m\), where \(m\) and \(k\) are two positive integers.

Let \(G\) be a graph with a maximum matching of size \(q\), and let \(p \leq q\) be a positive integer. Then \(G\) is called \((p, q)\)-extendable if every set of \(p\) independent edges can be extended to a matching of size \(q\). If \(G\) is a graph of even order \(n\) and \(n = 2q\), then \((p,q)\)-extendable graphs are exactly the \(p\)-extendable graphs defined by Plummer \([11]\) in \(1980\).

Let \(d \geq 3\) be an integer, and let \(G\) be a \(d\)-regular graph of order \(n\) with a maximum matching of size \(q = \frac{n-t}{2}\geq 3\) for an integer \(t \geq 1\) such that \(n – t\) is even. In this work, we prove that if

(i) \(n \leq {(t+4)(d+1)-5}\) or

(ii) \(n \leq (t+4)(d+2) – 1\) when \(d\) is odd,

then \(G\) is \((2, q)\)-extendable.

A graph with vertex set \(V\) is said to have a prime cordial labeling if there is a bijection \(f\) from \(V\) to \(\{1,2,\ldots,|V|\}\) such that if each edge \(uv\) is assigned the label \(1\) for the greatest common divisor \(\gcd(f(u), f(v)) = 1\) and \(0\) for \(\gcd(f(u), f(v)) = 1\), then the number of edges labeled with \(0\) and the number of edges labeled with \(1\) differ by at most \(1\). In this paper, we show that the Flower Snark and its related graphs are prime cordial for all \(n \geq 3\).

In [2] it is proved that if \(X = Cay(G, S)\) is a connected tetravalent Cayley graph on a regular \(p\)-group \(G\) (for \(p \neq 2, 5\)), then the right regular representation of \(G\) is normal in the automorphism group of \(X\). In this paper, we prove that a similar result holds, for \(p = 5\), under a slightly stronger hypothesis. Some remarkable examples are presented.

In this paper, we define, for a graph invariant \(\psi\), the deck ratio of \(\psi\) by \(D_\psi(G) = \frac{\psi(G)}{\Sigma_{v\in V(G)}\psi(G-v)}\). We give generic upper and lower bounds on \(D_\psi\) for monotone increasing and monotone decreasing invariants \(\psi\), respectively.

Then, we proceed to consider the Wiener index \(W(G)\), showing that \(D_W(G) \leq \frac{1}{|V(G)|-2}\). We show that equality is attained for a graph \(G\) if and only if every induced \(P_3\) subgraph of \(G\) is contained in a \(C_4\) subgraph. Such graphs have been previously studied under the name of self-repairing graphs.

We show that a graph on \(n \geq 4\) vertices with at least \(\frac{n^2-3n+6}{2} – n + 3\) edges is necessarily a self-repairing graph and that this is the best possible result. We also show that a \(2\)-connected graph is self-repairing if and only if all factors in its Cartesian product decomposition are.

Finally, some open problems about the deck ratio and about self-repairing graphs are posed at the end of the paper.

For a graph \(X\) and a digraph \(D\), we define the \(\beta\) transformation of \(X\) and the \(\alpha\) transformation of \(D\) denoted by \(X^\beta\) and \(D^\alpha\), respectively.\(D^\alpha\) is defined as the bipartite graph with vertex set \(V(D) \times \{0,1\}\) and edge set \(\{((v_i,0), (v_j, 1)) \mid v_i v_j \in A(D)\}\).\(X^\beta\) is defined as the bipartite graph with vertex set \(V(X) \times \{0,1\}\) and edge set \(\{((v_i,0), (v_j, 1)) \mid v_i v_j \in A(X)\}\), where \(X\) is the associated digraph of \(X\).In this paper, we give the relation between the eigenvalues of the digraph \(D\) and the graph \(D^\alpha\) when the adjacency matrix of \(D\) is normal. Especially, we obtain the eigenvalues of \(D^\alpha\) when \(D\) is some special Cayley digraph.

To study orthogonal arrays and signed orthogonal arrays, Ray-Chaudhuri and Singhi (\(1988\) and \(1994\)) considered some module spaces. Here, using a linear algebraic approach, we define an inclusion matrix and find its rank. In the special case of Latin squares, we show that there is a straightforward algorithm for generating a basis for this matrix using the so-called intercalates. We also extend this last idea.

Integral circulant graphs have been proposed as potential candidates for modelling quantum spin networks with perfect state transfer between antipodal sites in the network. We show that the diameter of these graphs is at most \(O(\ln \ln n)\), and further improve the recent result of Saxena, Severini, and Shparlinski.