We recall from [13] a shell graph of size $n$, denoted $C(n,n-3)$, is the graph obtained from the cycle $C_n(v_1,v_2,\dots ,v_{n-1})$ by adding $n-3$ consecutive chords incident at a common vertex, say $v_0$. The vertex $v_0$ of $C(n,n-3)$ is called the apex of the shell $C(n,n-3)$. The vertex $v_1$ of $C(n,n-3)$ is said to be at level 1.

A graph $C(2n,n-2)$ is called an alternate shell, if $C(2n,n-2)$ is obtained from the cycle $C_{2n}(v_0,v_1,v_2,\dots ,v_{2n-1})$ by adding $n-2$ chords between the vertex $v_0$ and the vertices $v_{2i+1}$, for $1\le i\le n-2$. If the vertex $v_i$ of $C(2n,n-2)$ at level 1 is adjacent with $v_0$, then $v_1$ is said to be at level 1 with a chord, otherwise the vertex $v_1$ is said to be at level 1 without a chord.

In 2009, Akelbek and Kirkland introduced a useful parameter called the scrambling index of a primitive digraph $D$, which is the smallest positive integer $k$ such that for every pair of vertices $u$ and $v$, there is a vertex $w$ such that we can get to $w$ from $u$ and $v$ in $D$ by walks of length $k$. In this paper, we study and obtain the scrambling indices of all primitive digraphs with exactly two cycles.

Given a tournament $T=(V,A)$, a subset $X$ of $V$ is an interval of $T$ provided that for every $a,b\in X$ and $x\in V–X$, $(a,x)\in A$ if and only if $(b,x)\in A$. For example, $\mathrm{\varnothing }$, $\{x\}$ ($x\in V$), and $V$ are intervals of $T$, called trivial intervals. A tournament, all the intervals of which are trivial, is indecomposable; otherwise, it is decomposable. A critical tournament is an indecomposable tournament $T$ of cardinality $\ge 5$ such that for any vertex $x$ of $T$, the tournament $T–x$ is decomposable. The critical tournaments are of odd cardinality and for all $n\ge 2$ there are exactly three critical tournaments on $2n+1$ vertices denoted by $T_{2n+1}$, $U_{2n+1}$, and $W_{2n+1}$. The tournaments $T_5$, $U_5$, and $W_5$ are the unique indecomposable tournaments on 5 vertices. We say that a tournament $T$ embeds into a tournament $T^{\prime }$ when $T$ is isomorphic to a subtournament of $T^{\prime }$. A diamond is a tournament on 4 vertices admitting only one interval of cardinality 3. We prove the following theorem: if a diamond and $T_5$ embed into an indecomposable tournament $T$, then $W_5$ and $U_5$ embed into $T^{\prime }$. To conclude, we prove the following: given an indecomposable tournament $T$ with $|V(T)|\ge 7$, $T$ is critical if and only if only one of the tournaments $T_7$, $U_7$, or $W_7$ embeds into $T$.

Let $\lambda K_{m,n}$ be a complete bipartite multigraph with two partite sets having $m$ and $n$ vertices, respectively. A $K_{p,q}$-factorization of $\lambda K_{m,n}$ is a set of edge-disjoint $K_{p,q}$-factors of $\lambda K_{m,n}$ which is a partition of the set of edges of $\lambda K_{m,n}$. When $\lambda =1$, Martin, in paper [Complete bipartite factorisations by complete bipartite graphs, Discrete Math., $167/168(1997),461-480]$, gave simple necessary conditions for such a factorization to exist, and conjectured those conditions are always sufficient. In this paper, we will give similar necessary conditions for $\lambda K_{m,n}$ to have a $K_{p,q}$-factorization, and prove the necessary conditions are always sufficient in many cases.

In this paper, we determine upper and lower bounds for the number of independent sets in a bicyclic graph in terms of its order. This
gives an upper bound for the total number of independent sets in a connected graph which contains at least two cycles. In each case, we characterize the extremal graphs.

Let $G$ be a connected graph of order $n$. Denote $p_u(G)$ the order of a longest path starting at vertex $u$ in $G$. In this paper, we prove that if $G$ has more than $t(\genfrac{}{}{0ex}{}{k}{2})+(\genfrac{}{}{0ex}{}{p+1}{2})+(n-k-1)$ edges, where $k\ge 2$, $n=t(k-1)+p+1$, $t\ge 0$ and $0\le p\le k-1$, then $p_u(G)>k$ for each vertex $u$ in $G$. By this result, we give an alternative proof of a result obtained by P. Wang et al. that if $G$ is a 2-connected graph on $n$ vertices and with more than $t(\genfrac{}{}{0ex}{}{k-2}{2})+(\genfrac{}{}{0ex}{}{p}{2})+(2n–3)$ edges, where $k\ge 3$, $n-2=t(k-2)+p$, $t\ge 0$ and $0\le p\le k-2$, then each edge of $G$ lies on a cycle of order more than $k$.

In this paper, we give some identities involving the harmonic numbers and the inverses of binomial coefficients.

In this paper, a new efficient computational algorithm is presented for solving cyclic heptadiagonal linear systems based on using the heptadiagonal linear solver and Sherman–Morrison–Woodbury formula. The implementation of the algorithm using computer algebra systems (CAS) such as MAPLE and MATLAB is straightforward. Two numerical examples are presented for illustration.

Let $G$ be a graph, and let $a,b$, and $k$ be nonnegative integers with $0\le a\le b$. A graph $G$ is called an $(a,b,k)$-critical graph if after deleting any $k$ vertices of $G$, the remaining graph of $G$ has an $[a,b]$-factor. In this paper, we prove that if $\delta (G)\ge a+k$ and $\alpha (G)\le \frac{4b(\delta (G)-a+1-1)}{(a+1{)}^2}$, then $G$ is an $(a,b,k)$-critical graph. Furthermore, it is shown that the result in this paper is best possible in some sense.

A characterization of $B$-H-unretractive bipartite graphs is given. Based on this, it is proved that there is no bipartite graph with endotype $1(\mathrm{mod}4)$.

In a graph $G=(V,E)$, an independent set is a subset $I$ of $V(G)$ such that no two vertices in $I$ are adjacent. A maximum independent set is an independent set of maximum size. A connected graph (respectively, graph) $G$ with vertex set $V(G)$ is called a quasi-tree graph (respectively, quasi-forest graph), if there exists a vertex $x\in V(G)$ such that $G–x$ is a tree (respectively, forest). In this paper, we study the problem of determining the largest and the second largest numbers of maximum independent sets among all quasi-tree graphs and quasi-forest graphs. Extremal graphs achieving these values are also given.

The notions of sum labelling and sum number of graphs were introduced by F. Harary [1] in 1990. A mapping $f$ is called a sum labelling of a graph $G(V,E)$ if it is an injection from $V$ to a set of positive integers such that $uv\in E$ if and only if there exists a vertex $w\in V$ such that $f(w)=f(x)+f(y)$. In this case, $w$ is called a working vertex. If $f$ is a sum labelling of $G$ with $r$ isolated vertices, for some nonnegative integer $r$, and $G$ contains no working vertex, $f$ is defined as an exclusive sum labelling of the graph $G$ by M. Miller et al. in paper [2]. The least possible number $r$ of such isolated vertices is called the exclusive sum number of $G$, denoted by $ϵ(G)$. If $ϵ(G)=\mathrm{\Delta }(G)$, the labelling is called $\mathrm{\Delta }$-optimum exclusive sum labelling and the graph is said to be $\mathrm{\Delta }$-optimum summable, where $\mathrm{\Delta }=\mathrm{\Delta }(G)$ denotes the maximum degree of vertices in $G$. By using the notion of $\mathrm{\Delta }$-optimum forbidden subgraph of a graph, the exclusive sum numbers of crown $C_n\odot K_1$ and $(C_n\odot K_1)$ are given in this paper. Some $\mathrm{\Delta }$-optimum forbidden subgraphs of trees are studied, and we prove that for any integer $\mathrm{\Delta }\ge 3$, there exist trees not $\mathrm{\Delta }$-optimum summable. A nontrivial upper bound of the exclusive sum numbers of trees is also given in this paper.

In this paper we obtain the Fibonacci length of amalgamated free products having as factors dihedral groups.

In [11], Zhu, Li, and Deng introduced the definition of implicit degree of a vertex $v$, denoted by $\text{id}(v)$. In this paper, we consider implicit degrees and the hamiltonicity of graphs and obtain that:
If $G$ is a $2$-connected graph of order $n$ such that $\text{id}(u)+\text{id}(v)\ge n–1$ for each pair of vertices $u$ and $v$ at distance $2$, then $G$ is hamiltonian, with some exceptions.

Let $C_k$ denote a cycle of length $k$ and let $S_k$ denote a star with $k$ edges. For graphs $F$, $G$, and $H$, a $(G,H)$-multidecomposition of $F$ is a partition of the edge set of $F$ into copies of $G$ and copies of $H$ with at least one copy of $G$ and at least one copy of $H$. In this paper, necessary and sufficient conditions for the existence of the $(C_k,S_k)$-multidecomposition of a complete bipartite graph are given.

This paper investigates the number of boundary cubic inner-forest maps and presents some formulae for such maps with the size (number of edges) and the valency of the root-face as two parameters. Further, by duality, some corresponding results for rooted outer-planar maps are obtained. It is also an answer to the open problem in $[15]$ and corrects the result on boundary cubic inner-tree maps in $[15]$.

The following two theorems are proved:
A closed knight’s tour exists on all $m\times n$ boards wrapped onto a cylinder so that the $m$ rows go around the cylinder, with one square removed, with the exception of the following boards:

(a) $n$ is even,

(b) $m\in \{1,2\}$

(c) $m=4$ and the removed square is in row 2 or 3;

(d) $m\ge 5$, $n=1$, and the removed square is in row 2, 3, …, or $m-1$.

A closed knight’s tour exists on all $m\times n$ boards wrapped onto a torus with one square removed except boards with $m$ and $n$ both even and $1\times 1$,$1\times 2$ and $2\times 1$ boards.

An independent set $S$ of a connected graph $G$ is called a \emph{frame} if $G–S$ is connected. If $|S|=k$, then $S$ is called a \emph{k-frame}. We prove the following theorem.
Let $k\ge 2$ be an integer, $G$ be a connected graph with $V(G)=\{v_1,v_2,\dots ,v_n\}$, and ${\mathrm{deg}}_G(u)$ denote the degree of a vertex $u$. Suppose that for every $3$-frame $S=\{v_a,v_b,v_c\}$ such that $1\le a\le b\le c\le n$, ${\mathrm{deg}}_G(v_c)\le a$, ${\mathrm{deg}}_G(v_b)\le b-1$, and ${\mathrm{deg}}_G(v_c)\le c–2$, it holds that$${\mathrm{deg}}_G(v_a)+{\mathrm{deg}}_G(v_b)+{\mathrm{deg}}_G(v_c)–|N(v_a)\cap N(v_b)\cap N(v_c)|\ge |G|–k+1.$$ Then $G$ has a spanning tree with at most $k$-leaves. Moreover, the condition is sharp.
This theorem is a generalization of the results of E. Flandrin, H.A. Jung, and H. Li (Discrete Math. $90(1991),41-52)$ and of A. Kyaw (Australasian Journal of Combinatorics. $37(2007),3-10)$ for traceability.

In this paper, the estimations of maximum genus orientable embeddings of graphs are studied, and an exponential lower bound for such numbers is found. Moreover, such two extremal embeddings (i.e., the maximum genus orientable embedding of the current graph and the minimum genus orientable embedding of the complete graph) are sometimes closely related to each other. As applications, we estimate the number of minimum genus orientable embeddings for the complete graph by estimating the number of maximum genus orientable embeddings for the current graph.

In this article, we characterize for which finite commutative rings $R$, The zero-divisor graph $\mathrm{\Gamma }(R)$,The line graph $L(\mathrm{\Gamma }(R))$, The complement graph $\overline{\mathrm{\Gamma }(R)}$, and The line graph for the complement graph $L(\overline{\mathrm{\Gamma }(R)})$.

The energy of a graph $G$ is defined as the sum of the absolute values of all the eigenvalues of the graph. In this paper, we consider the energy of the $3$-circulant graphs, and obtain a computation formula, and establish new results for a certain class of circulant graphs. At the same time, we give a conjecture: The largest energy of circulant graphs relates with their components.

In this paper, we present two criteria for a sequence lying along a ray of a combinatorial triangle to be unimodal, and give a correct
proof for the result of Belbachir and Szalay on unimodal rays of the generalized Pascal’s triangle.

In this paper, we introduce the notion of derivation in lattice implication algebra, and consider the properties of derivations in lattice implication algebras. We give an equivalent condition to be a derivation of a lattice implication algebra. Also, we characterize the fixed set $Fi{x}_d(L)$ and $Kerd$ by derivations. Moreover, we prove that if $d$ is a derivation of a lattice implication algebra, every filter $F$ is $d$-invariant.

In this paper, we derive a family of identities on the arbitrary subscripted Fibonacci and Lucas numbers. Furthermore, we construct the tridiagonal and symmetric tridiagonal family of matrices whose determinants form any linear subsequence of the Fibonacci numbers and Lucas numbers. Thus, we give a generalization of the results presented in Nalli and Civciv [A. Nalli, H. Civciv, A generalization of tridiagonal matrix determinants, Fibonacci and Lucas numbers, Chaos, Solitons and Fractals $2009;40(1):355.61]$ and Cahill and Narayan [N. D. Cahill, D. A. Narayan, Fibonacci and Lucas numbers as tridiagonal matrix determinants, The Fibonacci Quarterly, $2004;42(1):216–221]$.

A maximal independent set is an independent set that is not a proper subset of any other independent set. A connected graph (respectively, graph) $G$ with vertex set $V(G)$ is called a quasi-tree graph (respectively, quasi-forest graph), if there exists a vertex $x\in V(G)$ such that $G–x$ is a tree (respectively, forest). In this paper, we determine the second largest numbers of maximal independent sets among all quasi-tree graphs and quasi-forest graphs. We also characterize those extremal graphs achieving these values.

For a poset $P=(X,{\le }_P)$, the strict-double-bound graph ($sDB$-graph $sDB(P)$) is the graph on $X$ for which vertices $u$ and $v$ of $sDB(P)$ are adjacent if and only if $u\ne v$ and there exist $x$ and $y$ in $X$ distinct from $u$ and $v$ such that $x{\le }_{P}y$ and $x{\le }_{P}v{\le }_{P}y$. The strict-double-bound number $\zeta (G)$ of a graph $G$ is defined as $min\{n;G\cup {\overline{K}}_n\text{ is a strict-double-bound graph}\}$.

We obtain that for a spider $S_{n,m}$ ($n,m>3$) and a ladder $L_n$ ($n\ge 4$), $\lceil 2\sqrt{nm}\rceil \le \zeta (S_{n,m})\le n+m$, $\zeta (S_{n,n})=2n$, and $\lceil 2\sqrt{3n+2}\rceil \le \zeta (L_n)\le 2n$.

We conjectured in $[3]$ that every biconnected cyclic graph is the one-dimensional skeleton of a regular cellulation of the $3$-sphere and proved it is true for planar and hamiltonian graphs. In this paper, we introduce the class of weakly split graphs and prove the conjecture is true for such class. Hamiltonian, split, complete $k$-partite, and matrogenic cyclic graphs are weakly split.

Let $(X,\mathcal{B})$ be a $\lambda$-fold $G$-decomposition and let $G_i$, $i=1,\dots ,\mu$, be nonisomorphic proper subgraphs of $G$ without isolated vertices. Put ${\mathcal{B}}_i=\{B_i|B\in \mathcal{B}\}$, where ${\mathcal{B}}_{\mathcal{i}}$ is a subgraph of $B$  isomorphic to $G_i$. A $\{G_1,G_2,\dots ,G_{\mu }\}$-metamorphosis of $(X,\mathcal{B})$ is a rearrangement, for each $i=1,\dots ,\mu$, of the edges of $\bigcup _{B\in B}(E(B)\setminus {\mathcal{B}}_i))$ into a family ${\mathcal{F}}_i$ of copies of $G_i$ with a leave $L_i$, such that $(X,{\mathcal{B}}_i\cup {\mathcal{F}}_i,L_i)$ is a maximum packing of $\lambda H$ with copies of $G_i$. In this paper, we give a complete answer to the existence problem of an $S_{\lambda }(2,4,7)$ having a $\{C_4,K_3+e\}$-metamorphosis.

For a positive integer $m$, where $1\le m\le n$, the $m$-competition index (generalized competition index) of a primitive digraph $D$ of order $n$ is the smallest positive integer $k$ such that for every pair of vertices $x$ and $y$, there exist $m$ distinct vertices $v_1,v_2,\dots ,v_m$ such that there exist walks of length $k$ from $x$ to $v_i$ and from $y$ to $v_i$ for $1\le i\le m$. In this paper, we study the generalized competition indices of symmetric primitive digraphs with loop. We determine the generalized competition index set and characterize completely the symmetric primitive digraphs in this class such that the generalized competition index is equal to the maximum value.

We give a new combinatorial bijection between a certain set of balanced modular tableaux of Gusein-Zade, Luengo, and Melle-Hernandez and $k$-ribbon shapes. In addition, we also use the Schensted algorithm for the rim hook tableaux of Stanton and White to write down an explicit generating function for these balanced modular tableaux.

A $(k;g)$-cage is a graph with the minimum order among all $k$-regular graphs with girth $g$. As a special family of graphs, $(k;g)$-cages have a number of interesting properties. In this paper, we investigate various properties of cages, e.g., connectivity, the density of shortest cycles, bricks and braces.

Let $G=(V,E)$ be a digraph with $n$ vertices and $m$ arcs without loops and multiarcs, $V=\{v_1,v_2,\dots ,v_n\}$. Denote the outdegree and average $2$-outdegree of the vertex $v_i$ by $d_i^{+}$ and $m_i^{+}$, respectively. Let $A(G)$ be the adjacency matrix and $D(G)=\text{diag}(d_1^{+},d_2^{+},\dots ,d_n^{+})$ be the diagonal matrix with outdegrees of the vertices of the digraph $G$. Then we call $Q(G)=D(G)+A(G)$ the signless Laplacian matrix of $G$. In this paper, we obtain some upper and lower bounds for the spectral radius of $Q(G)$, which is called the signless Laplacian spectral radius of $G$. We also show that some bounds involving outdegrees and the average $2$-outdegrees of the vertices of $G$ can be obtained from our bounds.

Lee and Kong conjecture that if $n\ge 1$ is an odd number, then $St(a_1,a_0,\dots ,a_n)$ would be super edge-magic, and meanwhile they proved that the following graphs are super edge-magic: $St(m,n)$ ($n=0\mathrm{mod}(m+1)$), $St(1,k,n)$ ($k=1,2$ or $n$), $St(2,k,n)$ ($k=2,3$), $St(1,1,k,n)$ ($k=2,3$), $St(k,2,2,n)$ ($k=1,2$). In this paper, the conjecture is further discussed and it is proved that $St(1,m,n)$, $St(3,m,m+1)$, $St(n,n+1,n+2)$ are super edge-magic, and under some conditions $St(a_1,a_2,\dots ,a_{2n+1})$; $St(a_1,a_2,\dots ,a_{4n+1})$, $St(a_1,a_2,\dots ,a_{4n+3})$ are also super edge-magic.

We determine all connected odd graceful graphs of order $\le 6$. We show that if $G$ is an odd graceful graph, then $G\cup K_{m,n}$ is odd graceful for all $m,n\ge 1$. We give an analogous statement to the graceful graphs statement, and we show that some families of graphs are odd graceful.

In this paper, we provide a method to obtain the lower bound on the number of distinct maximum genus embeddings of the complete bipartite graph $K_{n,n}$ ($n$ is an odd number), which, in some sense, improves the results of S. Stahl and H. Ren.

For positive integer $n$, let $f_3(n)$ be the least upper bound of the sums of the lengths of the sides of $n$ cubes packed into a unit cube $C$ in three dimensions in such a way that the smaller cubes have sides parallel to those of $C$. In this paper, we improve the lower bound of $f_3(n)$.

The transformation graph $G^{+--}$ of a graph $G$ is the graph with vertex set $V(G)\cup E(G)$, in which two vertices $u$ and $uv$ are joined by an edge if one of the following conditions holds: (i) $u,v\in V(G)$ and they are adjacent in $G$, (ii) $u,v\in E(G)$ and they are not adjacent in $G$, (iii) one of $u$ and $wv$ is in $V(G)$ while the other is in $E(G)$, and they are not incident in $G$. In this paper, for any graph $G$, we determine the independence number and the connectivity of $G^{+--}$. Furthermore, we show that for a graph $G$ with no isolated vertices, $G^{+--}$ is hamiltonian if and only if $G$ is not a star and $G\notin \{2K_2,K_2\}$.

We introduce quasi-almostmedian graphs as a natural nonbipartite generalization of almostmedian graphs. They are filling a gap between quasi-median graphs and quasi-semimedian graphs. We generalize some results of almostmedian graphs and deduce some results from a bigger class of quasi-semimedian graphs. The consequence of this is another characterization of almostmedian graphs as well as two new characterizations of quasi-median graphs.

In this note, we establish a convolution formula for Bernoulli polynomials in a new and brief way, and some known results are derived as a special case.

In this study, we define the generalized $k$-order Fibonacci matrix and the $n\times n$ generalized Pascal matrix ${\mathcal{F}}_n(GF)$ associated with generalized $\mathcal{F}$-nomial coefficients. We find the inverse of the generalized Pascal matrix ${\mathcal{F}}_n(GF)$ associated with generalized $\mathcal{F}$-nomial coefficients. In the last section, we factorize this matrix via the generalized $k$-order Fibonacci matrix and give illustrative examples for these factorizations.

The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Let $\mathcal{G}$ be the set of unicyclic graphs of order $n$ with girth $g$. For all integers $n$ and $g$ with $5\le g\le n–6$, we determine the first $|\frac{g}{2}|+3$ spectral radii of unicyclic graphs in the set ${\mathcal{U}}_n^g$.

In this paper, we consider labelings of graphs in which the label on an edge is the absolute value of the difference of its vertex labels. Such a labeling using $\{0,1,2,\dots ,k-1\}$ is called $k$-equitable if the number of vertices (resp. edges) labeled $i$ and the number of vertices (resp. edges) labeled $j$ differ by at most one and is called $k$-balanced if the number of vertices labeled $i$ and the number of edges labeled $j$ differ by at most one. We determine which graphs in certain families are $k$-equitable or $k$-balanced and we give also some necessary conditions on these two labelings.

The study of chromatically unique graphs has been drawing much attention and many results are surveyed in $[4,12,13]$. The notion of adjoint polynomials of graphs was first introduced and applied to the study of the chromaticity of the complements of the graphs by Liu $[17]$ (see also $[4]$). Two invariants for adjoint equivalent graphs that have been employed successfully to determine chromatic unique graphs were introduced by Liu $[17]$ and Dong et al. $[4]$ respectively. In the paper, we shall utilize, among other things, these two invariants to investigate the chromaticity of the complement of the tadpole graphs $C_n(P_m)$, the graph obtained from a path $P_m$ and a cycle $C_n$ by identifying a pendant vertex of the path with a vertex of the cycle. Let $\overline{G}$ stand for the complement of a graph $G$. We prove the following results:

1. The graph $\overline{C_{n-1}(P_2)}$ is chromatically unique if and only if $n\ne 5,7$.
2. Almost every $\overline{C_n(P_m)}$ is not chromatically unique, where $n\ge 4$ and $m\ge 2$.

An $L(2,1)$-labelling of a graph $G$ is a function $f$ from the vertex set $V(G)$ to the set of all nonnegative integers such that $|f(x)–f(y)|\ge 2$ if $d(x,y)=1$ and $|f(x)–f(y)|\ge 1$ if $d(x,y)=2$, where $d(x,y)$ denotes the distance between $x$ and $y$ in $G$. The $(2,1)$-labelling number $\lambda (G)$ of $G$ is the smallest number $k$ such that $G$ has an $L(2,1)$-labelling with $max\{f(v):v\in V(G)\}=k$. Griggs and Yeh conjecture that $\lambda (G)\le {\mathrm{\Delta }}^2$ for any simple graph with maximum degree $\mathrm{\Delta }\ge 2$. This article considers the graphs formed by the cartesian product of $n$ ($n\ge 2$ graphs. The new graph satisfies the above conjecture (with minor exceptions). Moreover, we generalize our results in [19].

In this study, we first define new sequences named $(s,t)$-Jacobsthal and $(s,t)$ Jacobsthal-Lucas sequences. After that, by using these sequences, we establish $(s,t)$-Jacobsthal and $(s,t)$ Jacobsthal-Lucas matrix sequences. Finally, we present some important relationships between these matrix sequences.

Several transformations about ${}_{\gamma }{F}_6(1)$-series are established by applying the modified Abel lemma on summation by parts. As a consequence, a reciprocal relation on balanced ${}_3{F}_2(1)$-series is derived, which may also be considered as a nonterminating extension of Saalschütz’s theorem (1891).